Abstract group 1024.bvh: $C_2^2\times C_4^2:C_{16}$

• Label: 1024.bvh
• Order: \( 2^{10} \)
• Exponent: \( 2^{4} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{9} \)
• $\card{Z(G)}$: \( 2^{8} \)
• $\card{\Aut(G)}$: \( 2^{27} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{25} \cdot 3 \)
• Perm deg.: not computed
• Trans deg.: $1024$
• Rank: $5$

Group information

Description:	$C_2^2\times C_4^2:C_{16}$
Order:	\(1024\)\(\medspace = 2^{10} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$C_2^9.C_2^6.C_2^6.D_6.C_2^4$, of order \(402653184\)\(\medspace = 2^{27} \cdot 3 \)
Composition factors:	$C_2$ x 10
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Group statistics

Order	1	2	4	8	16
Elements	1	31	224	256	512	1024
Conjugacy classes	1	31	160	192	256	640
Divisions	1	31	88	48	32	200
Autjugacy classes	1	6	9	6	1	23

Dimension	1	2	4	8
Irr. complex chars.	512	128	0	0	640
Irr. rational chars.	32	64	56	48	200

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$1024$
Rank:	$5$
Inequivalent generating 5-tuples:	$833280$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{16}=b^{4}=c^{4}=d^{2}=e^{2}=[a,b]=[a,d]=[a,e]=[b,c]=[b,d]=[b,e]=[c,d]=[c,e]=[d,e]=1, c^{a}=c^{3} \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 10 \\ 14 & 11 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 13 & 24 \\ 24 & 21 \end{array}\right), \left(\begin{array}{rr} 1 & 24 \\ 8 & 25 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_4$ $\, \times\, $ $(C_4:C_{16})$
Semidirect product:	$(C_2^2\times C_4^2)$ $\,\rtimes\,$ $C_{16}$	$C_4^2$ $\,\rtimes\,$ $(C_2^2\times C_{16})$	$C_4$ $\,\rtimes\,$ $(C_2^2\times C_4\times C_{16})$	$(C_2\times C_4^2)$ $\,\rtimes\,$ $(C_2\times C_{16})$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_4^3)$ . $C_8$	$C_4^3$ . $(C_2\times C_8)$	$C_4$ . $(C_4^3.C_4)$	$C_4^2$ . $(C_2^3.D_4)$	all 175

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{32}\Z)$.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} \times C_{16} $
Schur multiplier:	$C_{2}^{9} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 8789 subgroups in 6933 conjugacy classes, 5077 normal (42 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^3\times C_4\times C_8$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^3\times C_4\times C_{16}$
Frattini:	$\Phi \simeq$ $C_2^2\times C_8$	$G/\Phi \simeq$ $C_2^5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_4^2:C_{16}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^2\times C_4^2:C_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_4^2:C_{16}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1024: $C_2^2\times C_4^2:C_{16}$

Order 512: $C_2\times C_4\times C_4:C_{16}$ x 24, $C_2^2\times (C_2\times C_4).C_4^2$ x 2, $C_2^3\times C_4:C_{16}$ x 2, $C_2^3\times C_4\times C_{16}$ x 2, $C_2^2\times C_4^2\times C_8$

Order 256: $C_4^2:C_{16}$ x 64, $C_2^3.\OD_{32}$ x 56, $C_2^2\times C_4\times C_{16}$ x 40, $C_2^3.\OD_{32}$ x 24, $C_2\times C_4^2\times C_8$ x 24, $C_2^3\times C_4\times C_8$ x 6, $C_2^4\times C_{16}$ x 4, $C_2^2\times C_4^3$

Order 128: $C_4^2.C_8$ x 224, $C_2^2\times C_4\times C_8$ x 144, $C_2\times C_4\times C_{16}$ x 128, $C_2^3\times C_{16}$ x 88, $C_4^2\times C_8$ x 64, $C_2^2.\OD_{32}$ x 32, $C_2\times C_4^3$ x 24, $C_2^3\times C_4^2$ x 7, $C_2^4\times C_8$ x 4

Order 64: $C_2\times C_4\times C_8$ x 528, $C_2^2\times C_{16}$ x 336, $C_2^2\times C_4^2$ x 172, $C_4:C_{16}$ x 128, $C_2^3\times C_8$ x 104, $C_4^3$ x 64, $C_4\times C_{16}$ x 64, $C_2^4\times C_4$ x 7

Order 32: $C_2\times C_4^2$ x 640, $C_2^2\times C_8$ x 448, $C_4\times C_8$ x 288, $C_2\times C_{16}$ x 256, $C_2^3\times C_4$ x 186, $C_2^5$

Order 16: $C_2^2\times C_4$ x 812, $C_2\times C_8$ x 368, $C_4^2$ x 352, $C_{16}$ x 32, $C_2^4$ x 31

Order 8: $C_2\times C_4$ x 672, $C_2^3$ x 155, $C_8$ x 48

Order 4: $C_2^2$ x 155, $C_4$ x 88

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1024: $C_2^2\times C_4^2:C_{16}$

Order 512: $C_2^2\times (C_2\times C_4).C_4^2$, $C_2\times C_4\times C_4:C_{16}$, $C_2^3\times C_4:C_{16}$, $C_2^3\times C_4\times C_{16}$, $C_2^2\times C_4^2\times C_8$

Order 256: $C_2^3\times C_4\times C_8$ x 6, $C_2^3.\OD_{32}$ x 2, $C_2^2\times C_4\times C_{16}$ x 2, $C_2\times C_4^2\times C_8$ x 2, $C_4^2:C_{16}$, $C_2^4\times C_{16}$, $C_2^3.\OD_{32}$, $C_2^2\times C_4^3$

Order 128: $C_2^2\times C_4\times C_8$ x 16, $C_2^3\times C_4^2$ x 5, $C_2^4\times C_8$ x 4, $C_2^3\times C_{16}$ x 4, $C_2\times C_4^3$ x 2, $C_4^2.C_8$ x 2, $C_2\times C_4\times C_{16}$ x 2, $C_4^2\times C_8$ x 2, $C_2^2.\OD_{32}$

Order 64: $C_2\times C_4\times C_8$ x 19, $C_2^3\times C_8$ x 18, $C_2^2\times C_4^2$ x 16, $C_2^4\times C_4$ x 5, $C_2^2\times C_{16}$ x 5, $C_4^3$ x 2, $C_4:C_{16}$, $C_4\times C_{16}$

Order 32: $C_2^2\times C_8$ x 26, $C_2^3\times C_4$ x 22, $C_2\times C_4^2$ x 19, $C_4\times C_8$ x 10, $C_2\times C_{16}$ x 4, $C_2^5$

Order 16: $C_2^2\times C_4$ x 33, $C_2\times C_8$ x 22, $C_4^2$ x 11, $C_2^4$ x 6, $C_{16}$

Order 8: $C_2\times C_4$ x 28, $C_2^3$ x 11, $C_8$ x 6

Order 4: $C_2^2$ x 11, $C_4$ x 9

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1024: $C_2^2\times C_4^2:C_{16}$ ($C_1$)

Order 512: $C_2\times C_4\times C_4:C_{16}$ ($C_2$) x 24, $C_2^2\times (C_2\times C_4).C_4^2$ ($C_2$) x 2, $C_2^3\times C_4:C_{16}$ ($C_2$) x 2, $C_2^3\times C_4\times C_{16}$ ($C_2$) x 2, $C_2^2\times C_4^2\times C_8$ ($C_2$)

Order 256: $C_4^2:C_{16}$ ($C_2^2$) x 64, $C_2^3.\OD_{32}$ ($C_4$) x 32, $C_2^3.\OD_{32}$ ($C_2^2$) x 24, $C_2^3.\OD_{32}$ ($C_2^2$) x 24, $C_2^2\times C_4\times C_{16}$ ($C_2^2$) x 24, $C_2\times C_4^2\times C_8$ ($C_2^2$) x 12, $C_2\times C_4^2\times C_8$ ($C_4$) x 12, $C_2^4\times C_{16}$ ($C_2^2$) x 4, $C_2^3\times C_4\times C_8$ ($C_2^2$) x 3, $C_2^3\times C_4\times C_8$ ($C_4$) x 3, $C_2^2\times C_4^3$ ($C_4$)

Order 128: $C_4^2.C_8$ ($C_2\times C_4$) x 192, $C_2^2\times C_4\times C_8$ ($C_2\times C_4$) x 70, $C_4^2\times C_8$ ($C_2\times C_4$) x 48, $C_4^2.C_8$ ($C_2^3$) x 32, $C_2\times C_4\times C_{16}$ ($C_2^3$) x 32, $C_2^3\times C_{16}$ ($C_2\times C_4$) x 32, $C_2^2.\OD_{32}$ ($C_2^3$) x 32, $C_2^3\times C_{16}$ ($C_2^3$) x 24, $C_2^2\times C_4\times C_8$ ($C_2^3$) x 18, $C_4^2\times C_8$ ($C_2^3$) x 16, $C_2^2\times C_4\times C_8$ ($C_8$) x 16, $C_2\times C_4^3$ ($C_2\times C_4$) x 12, $C_2\times C_4^3$ ($C_8$) x 12, $C_2^2\times C_4\times C_8$ ($Q_8$) x 8, $C_2^2\times C_4\times C_8$ ($D_4$) x 8, $C_2^3\times C_4^2$ ($C_8$) x 4, $C_2^4\times C_8$ ($C_2\times C_4$) x 3, $C_2^3\times C_4^2$ ($C_2\times C_4$) x 3, $C_2^4\times C_8$ ($C_2^3$)

Order 64: $C_2\times C_4\times C_8$ ($C_2^2\times C_4$) x 132, $C_4:C_{16}$ ($C_2^2\times C_4$) x 128, $C_2\times C_4\times C_8$ ($C_2\times C_8$) x 96, $C_2^2\times C_{16}$ ($C_2^2\times C_4$) x 96, $C_2^2\times C_4^2$ ($C_2\times C_8$) x 74, $C_2\times C_4\times C_8$ ($C_4:C_4$) x 48, $C_2\times C_4\times C_8$ ($C_4^2$) x 48, $C_4^3$ ($C_2\times C_8$) x 48, $C_2^2\times C_4^2$ ($C_{16}$) x 32, $C_2^3\times C_8$ ($C_2^2\times C_4$) x 29, $C_2\times C_4\times C_8$ ($C_2\times Q_8$) x 24, $C_2\times C_4\times C_8$ ($C_2\times D_4$) x 24, $C_2^2\times C_4^2$ ($C_2^2\times C_4$) x 18, $C_4^3$ ($C_2^2\times C_4$) x 16, $C_2^3\times C_8$ ($C_2\times C_8$) x 16, $C_2^2\times C_{16}$ ($C_2^4$) x 16, $C_2\times C_4\times C_8$ ($C_2^4$) x 12, $C_2^3\times C_8$ ($\OD_{16}$) x 8, $C_2^3\times C_8$ ($C_4:C_4$) x 8, $C_2^3\times C_8$ ($C_4^2$) x 8, $C_2^3\times C_8$ ($D_4:C_2$) x 8, $C_2^2\times C_4^2$ ($\OD_{16}$) x 8, $C_2^2\times C_4^2$ ($C_4:C_4$) x 8, $C_2^2\times C_4^2$ ($C_4^2$) x 8, $C_2^4\times C_4$ ($C_2\times C_8$) x 6, $C_2^3\times C_8$ ($C_2\times Q_8$) x 4, $C_2^3\times C_8$ ($C_2\times D_4$) x 4, $C_2^3\times C_8$ ($C_2^4$) x 3, $C_2^4\times C_4$ ($C_2^2\times C_4$)

Order 32: $C_2\times C_4^2$ ($C_2\times C_{16}$) x 192, $C_2\times C_4^2$ ($C_2^2\times C_8$) x 124, $C_2^2\times C_8$ ($C_4:C_8$) x 64, $C_4\times C_8$ ($C_2^2\times C_8$) x 64, $C_2^3\times C_4$ ($C_2\times C_{16}$) x 48, $C_2^2\times C_8$ ($C_2^2\times C_8$) x 48, $C_4\times C_8$ ($C_2^2.D_4$) x 48, $C_4\times C_8$ ($C_2\times C_4^2$) x 48, $C_2\times C_4^2$ ($C_4\times C_8$) x 48, $C_2\times C_4^2$ ($C_4:C_8$) x 48, $C_2^3\times C_4$ ($C_2^2\times C_8$) x 43, $C_2^2\times C_8$ ($C_2^2.D_4$) x 36, $C_2^2\times C_8$ ($C_2\times C_4^2$) x 36, $C_2^2\times C_8$ ($C_4\times Q_8$) x 32, $C_2^2\times C_8$ ($C_4\times D_4$) x 32, $C_2\times C_{16}$ ($C_2^3\times C_4$) x 32, $C_2^2\times C_8$ ($C_2^3\times C_4$) x 27, $C_2^2\times C_8$ ($C_2\times \OD_{16}$) x 24, $C_2\times C_4^2$ ($C_2\times \OD_{16}$) x 24, $C_2\times C_4^2$ ($C_2^2.D_4$) x 24, $C_2\times C_4^2$ ($C_2\times C_4^2$) x 24, $C_2^3\times C_4$ ($C_4\times C_8$) x 16, $C_2^3\times C_4$ ($\OD_{32}$) x 16, $C_2^3\times C_4$ ($C_4:C_8$) x 16, $C_4\times C_8$ ($C_2^3\times C_4$) x 16, $C_2^2\times C_8$ ($D_4:C_2^2$) x 12, $C_2^2\times C_8$ ($C_4^2:C_2$) x 12, $C_2\times C_4^2$ ($C_2^3\times C_4$) x 12, $C_2^3\times C_4$ ($C_2\times \OD_{16}$) x 8, $C_4\times C_8$ ($C_2^2\times Q_8$) x 8, $C_4\times C_8$ ($C_2^2\times D_4$) x 8, $C_2^2\times C_8$ ($C_2^2\times Q_8$) x 6, $C_2^2\times C_8$ ($C_2^2\times D_4$) x 6, $C_2^3\times C_4$ ($C_4^2:C_2$) x 4, $C_2^3\times C_4$ ($C_2^2.D_4$) x 4, $C_2^3\times C_4$ ($C_2\times C_4^2$) x 4, $C_2^3\times C_4$ ($C_2^3\times C_4$) x 3, $C_2^5$ ($C_2^2\times C_8$), $C_2^2\times C_8$ ($C_2^5$)

Order 16: $C_2^2\times C_4$ ($C_2^2\times C_{16}$) x 144, $C_4^2$ ($C_2^2\times C_{16}$) x 128, $C_2^2\times C_4$ ($C_4:C_{16}$) x 128, $C_2\times C_8$ ($C_4^2.C_4$) x 96, $C_2^2\times C_4$ ($C_4^2.C_4$) x 76, $C_2^2\times C_4$ ($C_4\times C_{16}$) x 64, $C_2^2\times C_4$ ($C_2\times C_4\times C_8$) x 60, $C_2\times C_8$ ($C_4:C_4^2$) x 48, $C_4^2$ ($C_2\times C_4\times C_8$) x 48, $C_4^2$ ($C_4^2.C_4$) x 48, $C_2^2\times C_4$ ($C_2\times \OD_{32}$) x 48, $C_2^2\times C_4$ ($C_2^3\times C_8$) x 33, $C_2^2\times C_4$ ($C_4^2.C_4$) x 28, $C_2\times C_8$ ($C_4^2.C_2^2$) x 24, $C_2\times C_8$ ($C_4^2:C_2^2$) x 24, $C_2\times C_8$ ($C_2^3\times C_8$) x 16, $C_2^2\times C_4$ ($C_4\times \OD_{16}$) x 16, $C_2^2\times C_4$ ($C_4:C_4^2$) x 16, $C_2\times C_8$ ($C_2^3.D_4$) x 14, $C_2\times C_8$ ($C_2^2\times C_4^2$) x 14, $C_2^2\times C_4$ ($C_2^2\times \OD_{16}$) x 12, $C_2\times C_8$ ($C_2^2\times \OD_{16}$) x 8, $C_4^2$ ($C_2^2\times \OD_{16}$) x 8, $C_4^2$ ($C_2^3\times C_8$) x 8, $C_4^2$ ($C_2^3.D_4$) x 8, $C_4^2$ ($C_2^2\times C_4^2$) x 8, $C_2^4$ ($C_2\times \OD_{32}$) x 8, $C_2^4$ ($C_2^2\times C_{16}$) x 8, $C_2\times C_8$ ($C_4^2:C_2^2$) x 6, $C_2^2\times C_4$ ($C_4^2:C_2^2$) x 6, $C_2^2\times C_4$ ($C_2^3.D_4$) x 6, $C_2^2\times C_4$ ($C_2^2\times C_4^2$) x 6, $C_2^4$ ($C_2\times C_4\times C_8$) x 4, $C_2^4$ ($C_4^2.C_4$) x 4, $C_2^4$ ($C_4^2.C_4$) x 4, $C_2^4$ ($C_2^3\times C_8$) x 3, $C_2\times C_8$ ($D_4:C_2^3$) x 2, $C_2\times C_8$ ($C_2^4\times C_4$) x 2, $C_2\times C_8$ ($Q_8\times C_2^3$), $C_2\times C_8$ ($D_4\times C_2^3$), $C_2^2\times C_4$ ($C_2^4\times C_4$)

Order 8: $C_2\times C_4$ ($C_4^2.C_8$) x 192, $C_2\times C_4$ ($C_2\times C_4\times C_{16}$) x 96, $C_2\times C_4$ ($C_4^2:C_8$) x 48, $C_2\times C_4$ ($C_2^3\times C_{16}$) x 48, $C_2\times C_4$ ($C_4^2.C_2^3$) x 34, $C_2^3$ ($C_4^2.C_8$) x 32, $C_2^3$ ($C_4^2.C_8$) x 32, $C_2\times C_4$ ($C_2^2\times C_4\times C_8$) x 22, $C_2\times C_4$ ($C_4^2.C_2^3$) x 18, $C_2^3$ ($C_4\times \OD_{32}$) x 16, $C_2^3$ ($C_2\times C_4\times C_{16}$) x 16, $C_2^3$ ($C_4^2:C_8$) x 16, $C_2\times C_4$ ($C_2^2\times \OD_{32}$) x 16, $C_8$ ($C_4^2.C_2^3$) x 16, $C_2^3$ ($C_2^2\times \OD_{32}$) x 12, $C_2^3$ ($C_2^3\times C_{16}$) x 12, $C_2\times C_4$ ($C_2^3.C_4^2$) x 12, $C_2\times C_4$ ($C_2^3.C_2^4$) x 12, $C_8$ ($C_2^3.C_2^4$) x 12, $C_2^3$ ($C_4^2.C_2^3$) x 6, $C_2^3$ ($C_4^2.C_2^3$) x 6, $C_2^3$ ($C_2^2\times C_4\times C_8$) x 6, $C_2\times C_4$ ($C_2^3\times \OD_{16}$) x 2, $C_8$ ($C_4^2.C_2^3$) x 2, $C_8$ ($C_4^2:C_2^3$) x 2, $C_2^3$ ($C_2^4\times C_8$), $C_2\times C_4$ ($C_2^4\times C_8$), $C_2\times C_4$ ($C_4^2:C_2^3$), $C_2\times C_4$ ($C_2^4.D_4$), $C_2\times C_4$ ($C_2^3\times C_4^2$)

Order 4: $C_2^2$ ($C_4^2:C_{16}$) x 64, $C_4$ ($C_2^3.\OD_{32}$) x 32, $C_2^2$ ($(C_2^2\times C_4):C_{16}$) x 24, $C_2^2$ ($C_2^3.\OD_{32}$) x 24, $C_4$ ($C_2^2\times C_4\times C_{16}$) x 16, $C_2^2$ ($C_2\times C_4\times \OD_{32}$) x 12, $C_2^2$ ($C_2^2\times C_4\times C_{16}$) x 12, $C_2^2$ ($C_4^3.C_4$) x 12, $C_4$ ($C_4^3.C_4$) x 12, $C_2^2$ ($C_2^3\times \OD_{32}$) x 2, $C_2^2$ ($C_2^4\times C_{16}$) x 2, $C_2^2$ ($C_4^2.C_2^4$), $C_2^2$ ($C_4^2.C_2^4$), $C_2^2$ ($C_2^3\times C_4\times C_8$), $C_4$ ($C_4^2.C_2^4$), $C_4$ ($C_4^2.C_2^4$), $C_4$ ($C_4^2.C_2^4$), $C_4$ ($C_2^3.C_2^5$)

Order 2: $C_2$ ($C_2\times C_4\times C_4:C_{16}$) x 24, $C_2$ ($C_2^2\times (C_4\times C_8).C_2^2$) x 2, $C_2$ ($C_2^3\times C_4:C_{16}$) x 2, $C_2$ ($C_2^2\times C_4\times C_4:C_8$), $C_2$ ($C_2^3\times C_4\times C_{16}$), $C_2$ ($C_2^2\times C_4\times \OD_{32}$)

Order 1: $C_1$ ($C_2^2\times C_4^2:C_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1024: $C_2^2\times C_4^2:C_{16}$ ($C_1$)

Order 512: $C_2^2\times (C_2\times C_4).C_4^2$ ($C_2$), $C_2\times C_4\times C_4:C_{16}$ ($C_2$), $C_2^3\times C_4:C_{16}$ ($C_2$), $C_2^3\times C_4\times C_{16}$ ($C_2$), $C_2^2\times C_4^2\times C_8$ ($C_2$)

Order 256: $C_2^3\times C_4\times C_8$ ($C_2^2$) x 3, $C_2^3\times C_4\times C_8$ ($C_4$) x 3, $C_4^2:C_{16}$ ($C_2^2$), $C_2^4\times C_{16}$ ($C_2^2$), $C_2^3.\OD_{32}$ ($C_2^2$), $C_2^2\times C_4^3$ ($C_4$), $C_2^3.\OD_{32}$ ($C_2^2$), $C_2^3.\OD_{32}$ ($C_4$), $C_2^2\times C_4\times C_{16}$ ($C_2^2$), $C_2\times C_4^2\times C_8$ ($C_2^2$), $C_2\times C_4^2\times C_8$ ($C_4$)

Order 128: $C_2^2\times C_4\times C_8$ ($C_2\times C_4$) x 7, $C_2^4\times C_8$ ($C_2\times C_4$) x 3, $C_2^3\times C_4^2$ ($C_2\times C_4$) x 3, $C_2^2\times C_4\times C_8$ ($C_2^3$) x 3, $C_2^3\times C_4^2$ ($C_8$) x 2, $C_2\times C_4^3$ ($C_2\times C_4$), $C_2\times C_4^3$ ($C_8$), $C_4^2.C_8$ ($C_2^3$), $C_4^2.C_8$ ($C_2\times C_4$), $C_2\times C_4\times C_{16}$ ($C_2^3$), $C_4^2\times C_8$ ($C_2^3$), $C_4^2\times C_8$ ($C_2\times C_4$), $C_2^4\times C_8$ ($C_2^3$), $C_2^3\times C_{16}$ ($C_2^3$), $C_2^3\times C_{16}$ ($C_2\times C_4$), $C_2^2\times C_4\times C_8$ ($Q_8$), $C_2^2\times C_4\times C_8$ ($D_4$), $C_2^2\times C_4\times C_8$ ($C_8$), $C_2^2.\OD_{32}$ ($C_2^3$)

Order 64: $C_2\times C_4\times C_8$ ($C_2^2\times C_4$) x 7, $C_2^2\times C_4^2$ ($C_2\times C_8$) x 6, $C_2^3\times C_8$ ($C_2^2\times C_4$) x 5, $C_2^4\times C_4$ ($C_2\times C_8$) x 4, $C_2\times C_4\times C_8$ ($C_2^4$) x 3, $C_2^2\times C_4^2$ ($C_2^2\times C_4$) x 3, $C_2^3\times C_8$ ($C_2\times C_8$) x 2, $C_2\times C_4\times C_8$ ($C_2\times C_8$), $C_2\times C_4\times C_8$ ($C_4:C_4$), $C_2\times C_4\times C_8$ ($C_4^2$), $C_2\times C_4\times C_8$ ($C_2\times Q_8$), $C_2\times C_4\times C_8$ ($C_2\times D_4$), $C_4^3$ ($C_2\times C_8$), $C_4^3$ ($C_2^2\times C_4$), $C_4:C_{16}$ ($C_2^2\times C_4$), $C_2^4\times C_4$ ($C_2^2\times C_4$), $C_2^3\times C_8$ ($\OD_{16}$), $C_2^3\times C_8$ ($C_4:C_4$), $C_2^3\times C_8$ ($C_4^2$), $C_2^3\times C_8$ ($C_2^4$), $C_2^3\times C_8$ ($D_4:C_2$), $C_2^3\times C_8$ ($C_2\times Q_8$), $C_2^3\times C_8$ ($C_2\times D_4$), $C_2^2\times C_4^2$ ($\OD_{16}$), $C_2^2\times C_4^2$ ($C_4:C_4$), $C_2^2\times C_4^2$ ($C_4^2$), $C_2^2\times C_4^2$ ($C_{16}$), $C_2^2\times C_{16}$ ($C_2^4$), $C_2^2\times C_{16}$ ($C_2^2\times C_4$)

Order 32: $C_2^3\times C_4$ ($C_2^2\times C_8$) x 6, $C_2\times C_4^2$ ($C_2^2\times C_8$) x 6, $C_2^2\times C_8$ ($C_2^3\times C_4$) x 5, $C_2^3\times C_4$ ($C_2\times C_{16}$) x 3, $C_2\times C_4^2$ ($C_2^3\times C_4$) x 3, $C_2^3\times C_4$ ($C_2\times \OD_{16}$) x 2, $C_2^3\times C_4$ ($C_4\times C_8$) x 2, $C_2^2\times C_8$ ($C_2^2\times C_8$) x 2, $C_2^2\times C_8$ ($C_2^2.D_4$) x 2, $C_2^2\times C_8$ ($C_2\times C_4^2$) x 2, $C_2^5$ ($C_2^2\times C_8$), $C_2^3\times C_4$ ($C_2^3\times C_4$), $C_2^3\times C_4$ ($C_4^2:C_2$), $C_2^3\times C_4$ ($C_2^2.D_4$), $C_2^3\times C_4$ ($C_2\times C_4^2$), $C_2^3\times C_4$ ($\OD_{32}$), $C_2^3\times C_4$ ($C_4:C_8$), $C_2^2\times C_8$ ($C_2^5$), $C_2^2\times C_8$ ($D_4:C_2^2$), $C_2^2\times C_8$ ($C_2^2\times Q_8$), $C_2^2\times C_8$ ($C_2^2\times D_4$), $C_2^2\times C_8$ ($C_2\times \OD_{16}$), $C_2^2\times C_8$ ($C_4\times Q_8$), $C_2^2\times C_8$ ($C_4\times D_4$), $C_2^2\times C_8$ ($C_4^2:C_2$), $C_2^2\times C_8$ ($C_4:C_8$), $C_4\times C_8$ ($C_2^2\times Q_8$), $C_4\times C_8$ ($C_2^2\times D_4$), $C_4\times C_8$ ($C_2^3\times C_4$), $C_4\times C_8$ ($C_2^2\times C_8$), $C_4\times C_8$ ($C_2^2.D_4$), $C_4\times C_8$ ($C_2\times C_4^2$), $C_2\times C_4^2$ ($C_2\times \OD_{16}$), $C_2\times C_4^2$ ($C_4\times C_8$), $C_2\times C_4^2$ ($C_2^2.D_4$), $C_2\times C_4^2$ ($C_2\times C_4^2$), $C_2\times C_4^2$ ($C_2\times C_{16}$), $C_2\times C_4^2$ ($C_4:C_8$), $C_2\times C_{16}$ ($C_2^3\times C_4$)

Order 16: $C_2^2\times C_4$ ($C_2^3\times C_8$) x 6, $C_2^2\times C_4$ ($C_2\times C_4\times C_8$) x 3, $C_2^2\times C_4$ ($C_2^2\times C_{16}$) x 3, $C_2^2\times C_4$ ($C_4^2.C_4$) x 3, $C_2\times C_8$ ($C_2^3\times C_8$) x 2, $C_2\times C_8$ ($C_2^3.D_4$) x 2, $C_2\times C_8$ ($C_2^2\times C_4^2$) x 2, $C_2^2\times C_4$ ($C_2^2\times \OD_{16}$) x 2, $C_2^2\times C_4$ ($C_4^2.C_4$) x 2, $C_2\times C_8$ ($C_4:C_4^2$), $C_2\times C_8$ ($D_4:C_2^3$), $C_2\times C_8$ ($Q_8\times C_2^3$), $C_2\times C_8$ ($D_4\times C_2^3$), $C_2\times C_8$ ($C_2^4\times C_4$), $C_2\times C_8$ ($C_2^2\times \OD_{16}$), $C_2\times C_8$ ($C_4^2.C_2^2$), $C_2\times C_8$ ($C_4^2:C_2^2$), $C_2\times C_8$ ($C_4^2:C_2^2$), $C_2\times C_8$ ($C_4^2.C_4$), $C_4^2$ ($C_2\times C_4\times C_8$), $C_4^2$ ($C_2^2\times \OD_{16}$), $C_4^2$ ($C_2^3\times C_8$), $C_4^2$ ($C_2^3.D_4$), $C_4^2$ ($C_2^2\times C_4^2$), $C_4^2$ ($C_2^2\times C_{16}$), $C_4^2$ ($C_4^2.C_4$), $C_2^4$ ($C_2\times C_4\times C_8$), $C_2^4$ ($C_2^3\times C_8$), $C_2^4$ ($C_2\times \OD_{32}$), $C_2^4$ ($C_2^2\times C_{16}$), $C_2^4$ ($C_4^2.C_4$), $C_2^4$ ($C_4^2.C_4$), $C_2^2\times C_4$ ($C_4\times \OD_{16}$), $C_2^2\times C_4$ ($C_4:C_4^2$), $C_2^2\times C_4$ ($C_4:C_{16}$), $C_2^2\times C_4$ ($C_2^4\times C_4$), $C_2^2\times C_4$ ($C_4\times C_{16}$), $C_2^2\times C_4$ ($C_4^2:C_2^2$), $C_2^2\times C_4$ ($C_2^3.D_4$), $C_2^2\times C_4$ ($C_2^2\times C_4^2$), $C_2^2\times C_4$ ($C_2\times \OD_{32}$)

Order 8: $C_2\times C_4$ ($C_2^3\times C_{16}$) x 3, $C_2\times C_4$ ($C_4^2.C_2^3$) x 3, $C_2\times C_4$ ($C_2^2\times C_4\times C_8$) x 3, $C_2\times C_4$ ($C_2^3\times \OD_{16}$) x 2, $C_2\times C_4$ ($C_4^2.C_2^3$) x 2, $C_2^3$ ($C_4^2.C_8$), $C_2^3$ ($C_4^2.C_8$), $C_2^3$ ($C_4\times \OD_{32}$), $C_2^3$ ($C_2\times C_4\times C_{16}$), $C_2^3$ ($C_4^2:C_8$), $C_2^3$ ($C_2^4\times C_8$), $C_2^3$ ($C_2^2\times \OD_{32}$), $C_2^3$ ($C_2^3\times C_{16}$), $C_2^3$ ($C_4^2.C_2^3$), $C_2^3$ ($C_4^2.C_2^3$), $C_2^3$ ($C_2^2\times C_4\times C_8$), $C_2\times C_4$ ($C_4^2.C_8$), $C_2\times C_4$ ($C_2\times C_4\times C_{16}$), $C_2\times C_4$ ($C_4^2:C_8$), $C_2\times C_4$ ($C_2^4\times C_8$), $C_2\times C_4$ ($C_4^2:C_2^3$), $C_2\times C_4$ ($C_2^4.D_4$), $C_2\times C_4$ ($C_2^3\times C_4^2$), $C_2\times C_4$ ($C_2^2\times \OD_{32}$), $C_2\times C_4$ ($C_2^3.C_4^2$), $C_2\times C_4$ ($C_2^3.C_2^4$), $C_8$ ($C_4^2.C_2^3$), $C_8$ ($C_4^2:C_2^3$), $C_8$ ($C_4^2.C_2^3$), $C_8$ ($C_2^3.C_2^4$)

Order 4: $C_2^2$ ($C_4^2:C_{16}$), $C_2^2$ ($C_2^3\times \OD_{32}$), $C_2^2$ ($C_2^4\times C_{16}$), $C_2^2$ ($C_4^2.C_2^4$), $C_2^2$ ($C_4^2.C_2^4$), $C_2^2$ ($C_2^3\times C_4\times C_8$), $C_2^2$ ($(C_2^2\times C_4):C_{16}$), $C_2^2$ ($C_2^3.\OD_{32}$), $C_2^2$ ($C_2\times C_4\times \OD_{32}$), $C_2^2$ ($C_2^2\times C_4\times C_{16}$), $C_2^2$ ($C_4^3.C_4$), $C_4$ ($C_4^2.C_2^4$), $C_4$ ($C_4^2.C_2^4$), $C_4$ ($C_4^2.C_2^4$), $C_4$ ($C_2^3.C_2^5$), $C_4$ ($C_2^3.\OD_{32}$), $C_4$ ($C_2^2\times C_4\times C_{16}$), $C_4$ ($C_4^3.C_4$)

Order 2: $C_2$ ($C_2^2\times (C_4\times C_8).C_2^2$), $C_2$ ($C_2\times C_4\times C_4:C_{16}$), $C_2$ ($C_2^2\times C_4\times C_4:C_8$), $C_2$ ($C_2^3\times C_4:C_{16}$), $C_2$ ($C_2^3\times C_4\times C_{16}$), $C_2$ ($C_2^2\times C_4\times \OD_{32}$)

Order 1: $C_1$ ($C_2^2\times C_4^2:C_{16}$)

Series

Derived series

$C_2^2\times C_4^2:C_{16}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2^2\times C_4^2:C_{16}$

$\rhd$

$C_2^2\times C_4^2\times C_8$

$\rhd$

$C_2^3\times C_4\times C_8$

$\rhd$

$C_2^4\times C_8$

$\rhd$

$C_2^3\times C_8$

$\rhd$

$C_2^2\times C_8$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^2\times C_4^2:C_{16}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3\times C_4\times C_8$

$\lhd$

$C_2^2\times C_4^2:C_{16}$

Character theory

Complex character table

The $640 \times 640$ character table is not available for this group.

Rational character table

See the $200 \times 200$ rational character table (warning: may be slow to load).