Abstract group 1024.ob: $C_2^3\times D_{16}:C_4$

• Label: 1024.ob
• Order: \( 2^{10} \)
• Exponent: \( 2^{4} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{29} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{24} \cdot 3 \cdot 7 \)
• Perm deg.: not computed
• Trans deg.: $512$
• Rank: $6$

Group information

Description:	$C_2^3\times D_{16}:C_4$
Order:	\(1024\)\(\medspace = 2^{10} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$(C_2^6\times C_4).C_2^6.C_2^6.C_2^6.\PSL(2,7)$, of order \(11274289152\)\(\medspace = 2^{29} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 10
Nilpotency class:	$4$
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	287	352	128	256	1024
Conjugacy classes	1	63	80	48	64	256
Divisions	1	63	48	24	24	160
Autjugacy classes	1	4	5	4	2	16

Dimension	1	2	4	8
Irr. complex chars.	128	96	32	0	256
Irr. rational chars.	64	48	24	24	160

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$512$
Rank:	$6$
Inequivalent generating 6-tuples:	$29998080$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{4}=b^{2}=c^{16}=d^{2}=e^{2}=f^{2}=[a,d]= \!\cdots\! \rangle}$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 17 & 2 \\ 0 & 15 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 0 & 15 \end{array}\right), \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $(D_{16}:C_4)$
Semidirect product:	$(C_4\times D_8)$ $\,\rtimes\,$ $C_2^4$	$(D_8:C_4)$ $\,\rtimes\,$ $C_2^4$	$D_8$ $\,\rtimes\,$ $(C_2^4\times C_4)$	$(C_2^3\times D_{16})$ $\,\rtimes\,$ $C_4$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^4\times C_8)$ . $D_4$	$(C_2^4\times C_4)$ . $D_8$	$C_2^5$ . $(C_2\times D_8)$	$(C_2\times D_8)$ . $C_2^5$	all 79

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

Homology

Abelianization:	$C_{2}^{5} \times C_{4} $
Schur multiplier:	$C_{2}^{15}$
Commutator length:	$1$

Subgroups

There are 64181 subgroups in 19317 conjugacy classes, 6709 normal (21 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^5$	$G/Z \simeq$ $C_2\times D_8$
Commutator:	$G' \simeq$ $C_8$	$G/G' \simeq$ $C_2^5\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_2^6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times D_{16}:C_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^3\times D_{16}:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2\times D_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times D_{16}:C_4$

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: $C_2^3\times D_{16}:C_4$

Order 512: $C_2^2\times C_4.(C_4\times D_4)$ x 56, $C_2^3\times C_4\times D_8$ x 2, $C_2^3\times C_2^2.D_8$ x 2, $C_2^3\times C_2.\OD_{32}$, $C_2^4\times D_{16}$, $C_2^3\times C_2.\SD_{32}$

Order 256: $C_4^2.D_8$ x 448, $C_4^2.C_2^4$ x 84, $C_2^3\times D_{16}$ x 60, $C_2^3.D_{16}$ x 56, $C_2^4.D_8$ x 28, $C_2^3.\OD_{32}$ x 28, $D_8\times C_2^4$ x 2, $C_4^2:C_2^4$ x 2, $C_2^4\times C_{16}$ x 2, $C_2^4.D_8$ x 2, $C_2^4.D_8$ x 2, $C_2^3\times C_4\times C_8$

Order 128: $C_2^2\times D_{16}$ x 560, $C_4^2.D_4$ x 560, $D_{16}:C_4$ x 512, $C_2^2.D_{16}$ x 224, $C_2^3.D_8$ x 112, $C_8.C_4^2$ x 112, $C_2^3\times D_8$ x 90, $C_4^2:C_2^3$ x 84, $C_2^3.D_8$ x 56, $C_2^3\times C_{16}$ x 52, $C_2^3.D_8$ x 42, $C_2^2\times C_4\times C_8$ x 28, $D_4\times C_2^4$ x 2, $C_2^5\times C_4$ x 2, $C_2^4\times C_8$ x 2, $C_2^4.D_4$ x 2, $C_2^5:C_4$ x 2, $C_2^3\times C_4^2$

Order 64: $C_2\times D_{16}$ x 960, $C_2^2\times D_8$ x 700, $C_4\times D_8$ x 576, $C_4^2:C_2^2$ x 560, $C_2^2.D_8$ x 224, $C_2^2\times C_{16}$ x 224, $C_2^2.D_8$ x 140, $D_8:C_4$ x 128, $C_2\times C_4\times C_8$ x 112, $C_2^4\times C_4$ x 93, $D_4\times C_2^3$ x 90, $C_{16}:C_4$ x 64, $C_{16}:C_4$ x 64, $C_2^4:C_4$ x 56, $C_2^3\times C_8$ x 52, $C_2^3.D_4$ x 42, $C_2^2\times C_4^2$ x 28, $C_2^6$ x 2

Order 32: $C_2\times D_8$ x 1080, $C_2^3\times C_4$ x 790, $C_2^2\times D_4$ x 700, $C_4\times D_4$ x 576, $D_{16}$ x 256, $C_2^2\times C_8$ x 224, $C_2^3:C_4$ x 224, $C_2\times C_{16}$ x 184, $C_2^2.D_4$ x 140, $D_4:C_4$ x 128, $C_2\times C_4^2$ x 112, $C_2^5$ x 93, $C_8:C_4$ x 72, $C_4\times C_8$ x 64

Order 16: $C_2^2\times C_4$ x 1500, $C_2\times D_4$ x 1080, $C_2^4$ x 791, $D_8$ x 272, $C_2\times C_8$ x 184, $C_2^2:C_4$ x 128, $C_4:C_4$ x 72, $C_4^2$ x 64, $C_{16}$ x 24

Order 8: $C_2^3$ x 1515, $C_2\times C_4$ x 632, $D_4$ x 272, $C_8$ x 24

Order 4: $C_2^2$ x 667, $C_4$ x 48

Order 2: $C_2$ x 63

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1024: $C_2^3\times D_{16}:C_4$

Order 512: $C_2^3\times C_2.\OD_{32}$, $C_2^3\times C_4\times D_8$, $C_2^2\times C_4.(C_4\times D_4)$, $C_2^4\times D_{16}$, $C_2^3\times C_2^2.D_8$, $C_2^3\times C_2.\SD_{32}$

Order 256: $C_2^3\times D_{16}$ x 2, $C_2^4\times C_{16}$ x 2, $C_4^2.C_2^4$ x 2, $D_8\times C_2^4$, $C_4^2:C_2^4$, $C_2^4.D_8$, $C_2^4.D_8$, $C_2^3\times C_4\times C_8$, $C_4^2.D_8$, $C_2^4.D_8$, $C_2^3.D_{16}$, $C_2^3.\OD_{32}$

Order 128: $C_2^3\times D_8$ x 4, $C_2^3\times C_{16}$ x 4, $C_2^4\times C_8$ x 2, $C_4^2:C_2^3$ x 2, $C_2^2\times D_{16}$ x 2, $C_4^2.D_4$ x 2, $C_2^3.D_8$ x 2, $C_2^2\times C_4\times C_8$ x 2, $D_{16}:C_4$, $C_2^3.D_8$, $C_2^2.D_{16}$, $C_8.C_4^2$, $D_4\times C_2^4$, $C_2^5\times C_4$, $C_2^4.D_4$, $C_2^5:C_4$, $C_2^3\times C_4^2$, $C_2^3.D_8$

Order 64: $C_2^3\times C_8$ x 6, $C_2^4\times C_4$ x 5, $C_2^2\times D_8$ x 5, $D_4\times C_2^3$ x 4, $C_2^2\times C_{16}$ x 4, $C_2\times C_4\times C_8$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_2^2\times C_4^2$ x 2, $C_2\times D_{16}$ x 2, $C_4\times D_8$ x 2, $C_2^2.D_8$ x 2, $C_2^2.D_8$, $C_{16}:C_4$, $D_8:C_4$, $C_{16}:C_4$, $C_2^6$, $C_2^4:C_4$

Order 32: $C_2^3\times C_4$ x 10, $C_2^2\times C_8$ x 7, $C_2^5$ x 5, $C_2^2\times D_4$ x 5, $C_2\times D_8$ x 5, $C_2\times C_{16}$ x 4, $C_4\times C_8$ x 2, $C_4\times D_4$ x 2, $C_2^2.D_4$ x 2, $C_2\times C_4^2$ x 2, $C_8:C_4$ x 2, $D_4:C_4$, $C_2^3:C_4$, $D_{16}$

Order 16: $C_2^2\times C_4$ x 11, $C_2^4$ x 8, $C_2\times C_8$ x 7, $C_2\times D_4$ x 5, $D_8$ x 3, $C_4:C_4$ x 2, $C_4^2$ x 2, $C_{16}$ x 2, $C_2^2:C_4$

Order 8: $C_2\times C_4$ x 10, $C_2^3$ x 9, $C_8$ x 4, $D_4$ x 3

Order 4: $C_2^2$ x 8, $C_4$ x 5

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1024: $C_2^3\times D_{16}:C_4$ ($C_1$)

Order 512: $C_2^2\times C_4.(C_4\times D_4)$ ($C_2$) x 56, $C_2^3\times C_4\times D_8$ ($C_2$) x 2, $C_2^3\times C_2^2.D_8$ ($C_2$) x 2, $C_2^3\times C_2.\OD_{32}$ ($C_2$), $C_2^4\times D_{16}$ ($C_2$), $C_2^3\times C_2.\SD_{32}$ ($C_2$)

Order 256: $C_4^2.D_8$ ($C_2^2$) x 448, $C_4^2.C_2^4$ ($C_2^2$) x 56, $C_2^3.D_{16}$ ($C_2^2$) x 56, $C_2^3\times D_{16}$ ($C_4$) x 32, $C_2^3\times D_{16}$ ($C_2^2$) x 28, $C_2^4.D_8$ ($C_2^2$) x 28, $C_2^3.\OD_{32}$ ($C_2^2$) x 28, $D_8\times C_2^4$ ($C_2^2$) x 2, $C_2^4\times C_{16}$ ($C_2^2$) x 2, $C_2^4.D_8$ ($C_2^2$) x 2, $C_2^3\times C_4\times C_8$ ($C_2^2$)

Order 128: $D_{16}:C_4$ ($C_2^3$) x 512, $C_2^2\times D_{16}$ ($C_2\times C_4$) x 448, $C_2^2.D_{16}$ ($C_2^3$) x 224, $C_4^2.D_4$ ($C_2^3$) x 224, $C_2^3.D_8$ ($C_2^3$) x 112, $C_8.C_4^2$ ($C_2^3$) x 112, $C_2^2\times D_{16}$ ($C_2^3$) x 112, $C_2^3\times D_8$ ($C_2\times C_4$) x 32, $C_2^3\times D_8$ ($C_2^3$) x 28, $C_2^3\times C_{16}$ ($C_2^3$) x 28, $C_2^3.D_8$ ($C_2^3$) x 28, $C_2^3\times C_{16}$ ($C_2\times C_4$) x 16, $C_2^2\times C_4\times C_8$ ($C_2^3$) x 14, $C_2^2\times C_4\times C_8$ ($D_4$) x 14, $C_2^4\times C_8$ ($C_2^3$), $C_2^4\times C_8$ ($D_4$), $C_2^3\times C_4^2$ ($D_4$)

Order 64: $C_2\times D_{16}$ ($C_2^2\times C_4$) x 896, $C_2^2\times D_8$ ($C_2^2\times C_4$) x 224, $D_8:C_4$ ($C_2^4$) x 128, $C_4\times D_8$ ($C_2^4$) x 128, $C_2^2\times C_{16}$ ($C_2^2\times C_4$) x 112, $C_2\times C_4\times C_8$ ($C_2\times D_4$) x 84, $C_{16}:C_4$ ($C_2^4$) x 64, $C_{16}:C_4$ ($C_2^4$) x 64, $C_2\times D_{16}$ ($C_2^4$) x 64, $C_2^2\times D_8$ ($C_2^4$) x 56, $C_2^2\times C_{16}$ ($C_2^4$) x 56, $C_2^2.D_8$ ($C_2^4$) x 56, $C_2\times C_4\times C_8$ ($C_2^4$) x 28, $C_2^3\times C_8$ ($C_2\times D_4$) x 21, $C_2^2\times C_4^2$ ($D_8$) x 14, $C_2^2\times C_4^2$ ($C_2\times D_4$) x 14, $C_2^3\times C_8$ ($D_4:C_2$) x 8, $C_2^3\times C_8$ ($C_2^2\times C_4$) x 8, $C_2^3\times C_8$ ($C_2^4$) x 7, $C_2^4\times C_4$ ($D_8$) x 2, $C_2^4\times C_4$ ($C_2\times D_4$)

Order 32: $D_{16}$ ($C_2^3\times C_4$) x 256, $C_2\times D_8$ ($C_2^3\times C_4$) x 224, $C_2\times C_{16}$ ($C_2^3\times C_4$) x 112, $C_2\times C_4^2$ ($C_2\times D_8$) x 84, $C_2^2\times C_8$ ($C_4\times D_4$) x 56, $C_4\times C_8$ ($C_2^2\times D_4$) x 56, $C_2^2\times C_8$ ($C_2^2\times D_4$) x 49, $C_2^3\times C_4$ ($C_2\times D_8$) x 35, $C_2^2\times C_8$ ($D_4:C_2^2$) x 28, $C_2^2\times C_8$ ($C_2^3\times C_4$) x 28, $C_2\times C_4^2$ ($C_2^2\times D_4$) x 28, $C_2\times D_8$ ($C_2^5$) x 16, $C_2\times C_{16}$ ($C_2^5$) x 16, $C_8:C_4$ ($C_2^5$) x 16, $C_2^3\times C_4$ ($D_8:C_2$) x 8, $C_2^3\times C_4$ ($C_4\times D_4$) x 8, $C_4\times C_8$ ($C_2^5$) x 8, $C_2^3\times C_4$ ($C_2^2\times D_4$) x 7, $C_2^2\times C_8$ ($C_2^5$) x 7, $C_2^5$ ($C_2\times D_8$)

Order 16: $C_2\times C_8$ ($C_4^2:C_2^2$) x 84, $C_2^2\times C_4$ ($C_2^2\times D_8$) x 77, $C_4^2$ ($C_2^2\times D_8$) x 56, $C_2^2\times C_4$ ($C_4\times D_8$) x 56, $D_8$ ($C_2^4\times C_4$) x 32, $C_2^2\times C_4$ ($D_8:C_2^2$) x 28, $C_2^2\times C_4$ ($C_4^2:C_2^2$) x 28, $C_2^4$ ($D_{16}:C_2$) x 16, $C_{16}$ ($C_2^4\times C_4$) x 16, $C_2\times C_8$ ($D_4\times C_2^3$) x 15, $C_2\times C_8$ ($D_4:C_2^3$) x 14, $C_2\times C_8$ ($C_2^4\times C_4$) x 14, $C_4^2$ ($D_4\times C_2^3$) x 8, $C_2^4$ ($C_4\times D_8$) x 8, $C_2^4$ ($C_2^2\times D_8$) x 7, $C_2^2\times C_4$ ($D_4\times C_2^3$) x 7, $C_2\times C_8$ ($C_2^6$)

Order 8: $C_2\times C_4$ ($C_4^2.D_4$) x 84, $C_2^3$ ($D_{16}:C_4$) x 64, $C_2^3$ ($C_{16}:C_2^3$) x 56, $C_2^3$ ($C_4^2.D_4$) x 28, $C_2\times C_4$ ($C_2^3\times D_8$) x 23, $C_2\times C_4$ ($D_8:C_2^3$) x 14, $C_2\times C_4$ ($C_4^2:C_2^3$) x 14, $C_8$ ($C_4^2:C_2^3$) x 14, $C_2^3$ ($C_2^3\times D_8$) x 7, $C_2\times C_4$ ($D_4\times C_2^4$), $C_8$ ($D_4:C_2^4$), $C_8$ ($C_2^5\times C_4$)

Order 4: $C_2^2$ ($C_4^2.D_8$) x 112, $C_2^2$ ($C_{16}:C_2^4$) x 28, $C_2^2$ ($C_4^2.C_2^4$) x 14, $C_4$ ($C_4^2.C_2^4$) x 14, $C_2^2$ ($D_8\times C_2^4$), $C_4$ ($D_8:C_2^4$), $C_4$ ($C_4^2:C_2^4$)

Order 2: $C_2$ ($C_2^2\times C_4.(C_4\times D_4)$) x 28, $C_2$ ($C_2^3\times C_{16}:C_2^2$) x 2, $C_2$ ($C_2^3\times C_4\times D_8$)

Order 1: $C_1$ ($C_2^3\times D_{16}:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1024: $C_2^3\times D_{16}:C_4$ ($C_1$)

Order 512: $C_2^3\times C_2.\OD_{32}$ ($C_2$), $C_2^3\times C_4\times D_8$ ($C_2$), $C_2^2\times C_4.(C_4\times D_4)$ ($C_2$), $C_2^4\times D_{16}$ ($C_2$), $C_2^3\times C_2^2.D_8$ ($C_2$), $C_2^3\times C_2.\SD_{32}$ ($C_2$)

Order 256: $C_2^4\times C_{16}$ ($C_2^2$) x 2, $D_8\times C_2^4$ ($C_2^2$), $C_2^3\times D_{16}$ ($C_2^2$), $C_2^3\times D_{16}$ ($C_4$), $C_4^2.C_2^4$ ($C_2^2$), $C_2^4.D_8$ ($C_2^2$), $C_2^3\times C_4\times C_8$ ($C_2^2$), $C_4^2.D_8$ ($C_2^2$), $C_2^4.D_8$ ($C_2^2$), $C_2^3.D_{16}$ ($C_2^2$), $C_2^3.\OD_{32}$ ($C_2^2$)

Order 128: $C_2^3\times C_{16}$ ($C_2^3$) x 2, $D_{16}:C_4$ ($C_2^3$), $C_2^3.D_8$ ($C_2^3$), $C_2^2.D_{16}$ ($C_2^3$), $C_8.C_4^2$ ($C_2^3$), $C_2^3\times D_8$ ($C_2^3$), $C_2^3\times D_8$ ($C_2\times C_4$), $C_2^4\times C_8$ ($C_2^3$), $C_2^4\times C_8$ ($D_4$), $C_2^3\times C_4^2$ ($D_4$), $C_2^2\times D_{16}$ ($C_2^3$), $C_2^2\times D_{16}$ ($C_2\times C_4$), $C_2^3\times C_{16}$ ($C_2\times C_4$), $C_4^2.D_4$ ($C_2^3$), $C_2^3.D_8$ ($C_2^3$), $C_2^2\times C_4\times C_8$ ($C_2^3$), $C_2^2\times C_4\times C_8$ ($D_4$)

Order 64: $C_2^3\times C_8$ ($C_2\times D_4$) x 2, $C_2^2\times C_{16}$ ($C_2^4$) x 2, $C_2\times C_4\times C_8$ ($C_2^4$), $C_2\times C_4\times C_8$ ($C_2\times D_4$), $C_{16}:C_4$ ($C_2^4$), $D_8:C_4$ ($C_2^4$), $C_{16}:C_4$ ($C_2^4$), $C_2^4\times C_4$ ($D_8$), $C_2^4\times C_4$ ($C_2\times D_4$), $C_2^2\times D_8$ ($C_2^4$), $C_2^2\times D_8$ ($C_2^2\times C_4$), $C_2^3\times C_8$ ($C_2^4$), $C_2^3\times C_8$ ($D_4:C_2$), $C_2^3\times C_8$ ($C_2^2\times C_4$), $C_2^2\times C_4^2$ ($D_8$), $C_2^2\times C_4^2$ ($C_2\times D_4$), $C_2\times D_{16}$ ($C_2^4$), $C_2\times D_{16}$ ($C_2^2\times C_4$), $C_2^2\times C_{16}$ ($C_2^2\times C_4$), $C_4\times D_8$ ($C_2^4$), $C_2^2.D_8$ ($C_2^4$)

Order 32: $C_2^3\times C_4$ ($C_2\times D_8$) x 2, $C_2^2\times C_8$ ($C_2^2\times D_4$) x 2, $C_2\times C_{16}$ ($C_2^5$) x 2, $C_2^5$ ($C_2\times D_8$), $C_2^3\times C_4$ ($C_2^2\times D_4$), $C_2^3\times C_4$ ($D_8:C_2$), $C_2^3\times C_4$ ($C_4\times D_4$), $C_2\times D_8$ ($C_2^5$), $C_2\times D_8$ ($C_2^3\times C_4$), $C_2^2\times C_8$ ($C_2^5$), $C_2^2\times C_8$ ($D_4:C_2^2$), $C_2^2\times C_8$ ($C_2^3\times C_4$), $C_2^2\times C_8$ ($C_4\times D_4$), $C_4\times C_8$ ($C_2^5$), $C_4\times C_8$ ($C_2^2\times D_4$), $C_2\times C_4^2$ ($C_2^2\times D_4$), $C_2\times C_4^2$ ($C_2\times D_8$), $D_{16}$ ($C_2^3\times C_4$), $C_2\times C_{16}$ ($C_2^3\times C_4$), $C_8:C_4$ ($C_2^5$)

Order 16: $C_2\times C_8$ ($D_4\times C_2^3$) x 2, $C_2^2\times C_4$ ($C_2^2\times D_8$) x 2, $D_8$ ($C_2^4\times C_4$), $C_2\times C_8$ ($C_2^6$), $C_2\times C_8$ ($D_4:C_2^3$), $C_2\times C_8$ ($C_2^4\times C_4$), $C_2\times C_8$ ($C_4^2:C_2^2$), $C_4^2$ ($D_4\times C_2^3$), $C_4^2$ ($C_2^2\times D_8$), $C_2^4$ ($C_2^2\times D_8$), $C_2^4$ ($D_{16}:C_2$), $C_2^4$ ($C_4\times D_8$), $C_2^2\times C_4$ ($D_4\times C_2^3$), $C_2^2\times C_4$ ($D_8:C_2^2$), $C_2^2\times C_4$ ($C_4^2:C_2^2$), $C_2^2\times C_4$ ($C_4\times D_8$), $C_{16}$ ($C_2^4\times C_4$)

Order 8: $C_2\times C_4$ ($C_2^3\times D_8$) x 2, $C_2^3$ ($D_{16}:C_4$), $C_2^3$ ($C_2^3\times D_8$), $C_2^3$ ($C_{16}:C_2^3$), $C_2^3$ ($C_4^2.D_4$), $C_2\times C_4$ ($D_4\times C_2^4$), $C_2\times C_4$ ($D_8:C_2^3$), $C_2\times C_4$ ($C_4^2:C_2^3$), $C_2\times C_4$ ($C_4^2.D_4$), $C_8$ ($D_4:C_2^4$), $C_8$ ($C_2^5\times C_4$), $C_8$ ($C_4^2:C_2^3$)

Order 4: $C_2^2$ ($D_8\times C_2^4$), $C_2^2$ ($C_{16}:C_2^4$), $C_2^2$ ($C_4^2.C_2^4$), $C_2^2$ ($C_4^2.D_8$), $C_4$ ($D_8:C_2^4$), $C_4$ ($C_4^2:C_2^4$), $C_4$ ($C_4^2.C_2^4$)

Order 2: $C_2$ ($C_2^3\times C_4\times D_8$), $C_2$ ($C_2^2\times C_4.(C_4\times D_4)$), $C_2$ ($C_2^3\times C_{16}:C_2^2$)

Order 1: $C_1$ ($C_2^3\times D_{16}:C_4$)

Series

Derived series

$C_2^3\times D_{16}:C_4$

$\rhd$

$C_8$

$\rhd$

$C_1$

Chief series

$C_2^3\times D_{16}:C_4$

$\rhd$

$C_2^4\times D_{16}$

$\rhd$

$C_2^4\times C_{16}$

$\rhd$

$C_2^4\times C_8$

$\rhd$

$C_2^3\times C_8$

$\rhd$

$C_2^2\times C_8$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3\times D_{16}:C_4$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^5$

$\lhd$

$C_2^3\times C_4^2$

$\lhd$

$C_2^3\times C_4\times C_8$

$\lhd$

$C_2^3\times D_{16}:C_4$

Character theory

Complex character table

See the $256 \times 256$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $160 \times 160$ rational character table (warning: may be slow to load).