Abstract group 512.54138: $(C_2\times C_4^2):C_{16}$

• Label: 512.54138
• Order: \( 2^{9} \)
• Exponent: \( 2^{4} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2^{6} \)
• $\card{\Aut(G)}$: \( 2^{18} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{15} \)
• Perm deg.: $28$
• Trans deg.: $512$
• Rank: $3$

Group information

Description:	$(C_2\times C_4^2):C_{16}$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$C_2^9.C_2^6.C_2^3$, of order \(262144\)\(\medspace = 2^{18} \)
Composition factors:	$C_2$ x 9
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	15	112	128	256	512
Conjugacy classes	1	15	64	80	64	224
Divisions	1	15	36	20	8	80
Autjugacy classes	1	8	9	5	1	24

Dimension	1	2	4	8
Irr. complex chars.	128	96	0	0	224
Irr. rational chars.	8	28	24	20	80

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$512$
Rank:	$3$
Inequivalent generating triples:	$168$

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:	$\langle a, b, c, d \mid a^{4}=b^{2}=c^{16}=d^{4}=[a,b]=[a,d]=[b,c]=[b,d]=1, c^{a}=bc, d^{c}=d^{3} \rangle$
Permutation group:	Degree $28$ $\langle(1,2,4,7,3,6,9,12,5,8,11,14,10,13,15,16)(22,24)(26,27), (1,3,5,10)(2,6,8,13) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4^2.C_8)$ $\,\rtimes\,$ $C_4$	$(C_2\times C_4^2)$ $\,\rtimes\,$ $C_{16}$	$C_4$ $\,\rtimes\,$ $(C_2^2.\OD_{32})$	$(C_2\times C_4)$ $\,\rtimes\,$ $(C_4:C_{16})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $(C_4^2.D_4)$	$(C_2\times C_4^3)$ . $C_4$	$C_4^2$ . $(C_8:C_4)$	$C_4^2$ . $(C_4:C_8)$	all 117

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 1306 subgroups in 846 conjugacy classes, 434 normal (46 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3\times C_8$	$G/Z \simeq$ $C_2^3$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2\times C_4\times C_{16}$
Frattini:	$\Phi \simeq$ $C_2^3\times C_8$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $(C_2\times C_4^2):C_{16}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_2\times C_4^2):C_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2\times C_4^2):C_{16}$

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

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

Order 128: $C_4^2.C_8$ x 16, $C_2^2.\OD_{32}$ x 8, $C_2^2\times C_4\times C_8$ x 6, $C_4^2\times C_8$ x 4, $C_2^3\times C_{16}$ x 4, $C_2\times C_4^3$

Order 64: $C_2\times C_4\times C_8$ x 48, $C_2^2\times C_{16}$ x 32, $C_4:C_{16}$ x 16, $C_2^2\times C_4^2$ x 7, $C_4^3$ x 4, $C_2^3\times C_8$ x 4

Order 32: $C_4\times C_8$ x 56, $C_2\times C_4^2$ x 56, $C_2^2\times C_8$ x 44, $C_2\times C_{16}$ x 40, $C_2^3\times C_4$ x 7

Order 16: $C_2^2\times C_4$ x 78, $C_2\times C_8$ x 76, $C_4^2$ x 64, $C_{16}$ x 8, $C_2^4$

Order 8: $C_2\times C_4$ x 136, $C_8$ x 20, $C_2^3$ x 15

Order 4: $C_4$ x 36, $C_2^2$ x 35

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 256: $C_2^3.\OD_{32}$, $C_2^3.\OD_{32}$, $C_2\times C_4^2\times C_8$

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

Order 64: $C_2\times C_4\times C_8$ x 14, $C_2^2\times C_{16}$ x 5, $C_2^2\times C_4^2$ x 4, $C_2^3\times C_8$ x 3, $C_4^3$ x 2, $C_4:C_{16}$

Order 32: $C_2^2\times C_8$ x 19, $C_2\times C_4^2$ x 17, $C_4\times C_8$ x 6, $C_2\times C_{16}$ x 5, $C_2^3\times C_4$ x 4

Order 16: $C_2^2\times C_4$ x 26, $C_2\times C_8$ x 23, $C_4^2$ x 10, $C_2^4$, $C_{16}$

Order 8: $C_2\times C_4$ x 34, $C_2^3$ x 8, $C_8$ x 5

Order 4: $C_2^2$ x 16, $C_4$ x 9

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $(C_2\times C_4^2):C_{16}$ ($C_1$)

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

Order 128: $C_4^2.C_8$ ($C_4$) x 8, $C_2^3\times C_{16}$ ($C_2^2$) x 4, $C_2^2\times C_4\times C_8$ ($C_2^2$) x 3, $C_2^2\times C_4\times C_8$ ($C_4$) x 3, $C_2\times C_4^3$ ($C_4$)

Order 64: $C_2\times C_4\times C_8$ ($D_4$) x 10, $C_2^2\times C_{16}$ ($C_2\times C_4$) x 8, $C_2\times C_4\times C_8$ ($Q_8$) x 6, $C_2\times C_4\times C_8$ ($C_2\times C_4$) x 4, $C_2\times C_4\times C_8$ ($C_8$) x 4, $C_2^2\times C_4^2$ ($C_8$) x 4, $C_2^3\times C_8$ ($C_2\times C_4$) x 3, $C_2^2\times C_4^2$ ($C_2\times C_4$) x 3, $C_2^3\times C_8$ ($C_2^3$)

Order 32: $C_4\times C_8$ ($C_4:C_4$) x 8, $C_4\times C_8$ ($C_2^2:C_4$) x 8, $C_2\times C_4^2$ ($\OD_{16}$) x 8, $C_2\times C_4^2$ ($C_{16}$) x 8, $C_2^3\times C_4$ ($C_2\times C_8$) x 6, $C_2^2\times C_8$ ($C_4:C_4$) x 6, $C_2\times C_4^2$ ($C_4:C_4$) x 6, $C_2^2\times C_8$ ($C_2\times D_4$) x 5, $C_2^2\times C_8$ ($\OD_{16}$) x 4, $C_2^2\times C_8$ ($C_2\times C_8$) x 4, $C_2^2\times C_8$ ($D_4:C_2$) x 4, $C_2^2\times C_8$ ($C_2\times Q_8$) x 3, $C_2^2\times C_8$ ($C_2^2:C_4$) x 2, $C_2^2\times C_8$ ($C_4^2$) x 2, $C_2^2\times C_8$ ($C_2^2\times C_4$) x 2, $C_2\times C_4^2$ ($C_2\times C_8$) x 2, $C_2\times C_4^2$ ($C_2^2:C_4$) x 2, $C_2\times C_4^2$ ($C_4^2$) x 2, $C_2^3\times C_4$ ($C_2^2\times C_4$)

Order 16: $C_2^2\times C_4$ ($\OD_{32}$) x 12, $C_2^2\times C_4$ ($C_2\times C_{16}$) x 12, $C_2^2\times C_4$ ($C_4:C_8$) x 12, $C_2\times C_8$ ($C_4:C_8$) x 8, $C_2^2\times C_4$ ($C_2\times \OD_{16}$) x 6, $C_2\times C_8$ ($C_2^2:Q_8$) x 4, $C_2\times C_8$ ($C_4:D_4$) x 4, $C_2\times C_8$ ($C_4\times Q_8$) x 4, $C_2\times C_8$ ($C_4\times D_4$) x 4, $C_2\times C_8$ ($C_2.C_4^2$) x 4, $C_4^2$ ($C_2^2:C_8$) x 4, $C_4^2$ ($C_8:C_4$) x 4, $C_4^2$ ($C_2.C_4^2$) x 4, $C_4^2$ ($C_4:C_8$) x 4, $C_2^2\times C_4$ ($C_2^2:C_8$) x 4, $C_2^2\times C_4$ ($C_4\times C_8$) x 4, $C_2\times C_8$ ($C_4:Q_8$) x 3, $C_2^2\times C_4$ ($C_2^2.D_4$) x 3, $C_2\times C_8$ ($C_4.Q_8$) x 2, $C_2\times C_8$ ($C_4^2:C_2$) x 2, $C_2\times C_8$ ($C_2^2.D_4$) x 2, $C_2\times C_8$ ($C_2^3:C_4$) x 2, $C_2^2\times C_4$ ($C_4^2:C_2$) x 2, $C_2\times C_8$ ($C_4:D_4$), $C_2^4$ ($C_2^2\times C_8$), $C_2^2\times C_4$ ($C_2^2\times C_8$), $C_2^2\times C_4$ ($C_2^3:C_4$), $C_2^2\times C_4$ ($C_2\times C_4^2$)

Order 8: $C_2\times C_4$ ($C_4:C_{16}$) x 24, $C_2\times C_4$ ($C_2^2:C_{16}$) x 8, $C_2\times C_4$ ($C_2^2.C_4^2$) x 8, $C_2^3$ ($C_2\times \OD_{32}$) x 6, $C_2\times C_4$ ($C_4:\OD_{16}$) x 6, $C_2\times C_4$ ($C_{16}:C_4$) x 4, $C_2\times C_4$ ($C_4\times C_{16}$) x 4, $C_2\times C_4$ ($C_4^2.C_4$) x 4, $C_8$ ($C_2^3.C_2^3$) x 4, $C_2^3$ ($C_4^2.C_4$) x 3, $C_2\times C_4$ ($C_4^2.C_4$) x 3, $C_2^3$ ($C_2^2\times C_{16}$) x 2, $C_2^3$ ($C_4^2.C_4$) x 2, $C_2\times C_4$ ($C_2^2:\OD_{16}$) x 2, $C_2\times C_4$ ($C_4\times \OD_{16}$) x 2, $C_2\times C_4$ ($C_2^3.D_4$) x 2, $C_2\times C_4$ ($C_4:C_4^2$) x 2, $C_2\times C_4$ ($C_4^2.C_4$) x 2, $C_8$ ($C_2^3.C_2^3$) x 2, $C_8$ ($C_2^3.C_2^3$) x 2, $C_2^3$ ($C_2^3:C_8$), $C_2^3$ ($C_2\times C_4\times C_8$), $C_2\times C_4$ ($C_2^3:C_8$), $C_2\times C_4$ ($C_4.C_4^2$), $C_2\times C_4$ ($C_4^2:C_4$), $C_2\times C_4$ ($C_4^2:C_4$), $C_2\times C_4$ ($C_2^2.C_4^2$)

Order 4: $C_4$ ($C_2^2.\OD_{32}$) x 8, $C_2^2$ ($C_4:\OD_{32}$) x 6, $C_2^2$ ($C_4^2.C_8$) x 6, $C_2^2$ ($C_4^2.C_8$) x 4, $C_2^2$ ($C_4^2.C_8$) x 4, $C_2^2$ ($C_2^2:\OD_{32}$) x 2, $C_2^2$ ($C_2^3:C_{16}$) x 2, $C_2^2$ ($C_4\times \OD_{32}$) x 2, $C_2^2$ ($C_4^2.D_4$) x 2, $C_2^2$ ($C_4^2:C_8$) x 2, $C_4$ ($C_4^2.D_4$) x 2, $C_4$ ($C_4^3.C_2$) x 2, $C_2^2$ ($C_8.C_4^2$), $C_2^2$ ($C_2\times C_4\times C_{16}$), $C_2^2$ ($C_4^2:C_8$), $C_2^2$ ($C_4^2:C_8$), $C_2^2$ ($C_2^3.C_4^2$), $C_4$ ($C_4^2:C_8$), $C_4$ ($C_4^2:C_8$), $C_4$ ($C_2^3.C_4^2$), $C_4$ ($C_2^4.C_2^3$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $(C_2\times C_4^2):C_{16}$ ($C_1$)

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

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

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

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

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

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

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

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

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

Series

Derived series

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

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$C_2^3.\OD_{32}$

$\rhd$

$C_2^2\times C_4\times C_8$

$\rhd$

$C_2^3\times C_8$

$\rhd$

$C_2^2\times C_8$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3\times C_8$

$\lhd$

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

Character theory

Complex character table

See the $224 \times 224$ 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 $80 \times 80$ rational character table.