Abstract group 512.1997: $(C_4\times C_8).C_4^2$

• Label: 512.1997
• Order: \( 2^{9} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{14} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $20$
• Trans deg.: $64$
• Rank: $2$

Group information

Description:	$(C_4\times C_8).C_4^2$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$(C_2\times C_4).C_2^6.C_2^5$, of order \(16384\)\(\medspace = 2^{14} \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$6$
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
Elements	1	71	344	96	512
Conjugacy classes	1	9	16	12	38
Divisions	1	9	11	6	27
Autjugacy classes	1	6	6	3	16

Dimension	1	2	4	8
Irr. complex chars.	16	4	14	4	38
Irr. rational chars.	4	10	4	9	27

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$64$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	9	10	10

Constructions

Presentation:	$\langle a, b, c, d \mid a^{4}=b^{8}=c^{2}=d^{8}=[a,c]=[b,c]=[c,d]=1, b^{a}=b^{7}d, d^{a}=b^{2}cd, d^{b}=d^{5} \rangle$
Permutation group:	Degree $20$ $\langle(1,2,3,8)(4,9,14,16)(5,10,12,7)(6,15,11,13)(17,18,20,19), (2,6,9,7,10,16,11,8) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4^2.Q_8)$ $\,\rtimes\,$ $C_4$	$(C_2^3.C_4^2)$ $\,\rtimes\,$ $C_4$	$((C_4\times C_8):C_4)$ $\,\rtimes\,$ $C_4$ (4)		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4:D_8)$ . $Q_8$	$(C_4:D_8)$ . $D_4$	$(C_4\times C_8)$ . $C_4^2$	$(C_4^2.D_4)$ . $C_2^2$	all 24

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

Homology

Abelianization:	$C_{4}^{2} $
Schur multiplier:	$C_{2}^{2} \times C_{8}$
Commutator length:	$1$

Subgroups

There are 1798 subgroups in 256 conjugacy classes, 38 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $(C_4\times C_8):C_4$
Commutator:	$G' \simeq$ $C_4\times C_8$	$G/G' \simeq$ $C_4^2$
Frattini:	$\Phi \simeq$ $C_4^2.D_4$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $(C_4\times C_8).C_4^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_4\times C_8).C_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $(C_4\times C_8):C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_4\times C_8).C_4^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 512: $(C_4\times C_8).C_4^2$

Order 256: $C_4^2.(C_2^2\times C_4)$ x 2, $(C_2\times C_8).D_8$

Order 128: $(C_4\times C_8):C_4$ x 6, $C_2^4.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.Q_8$, $C_4^2.D_4$, $C_2^3.C_4^2$

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

Order 32: $D_4:C_4$ x 8, $C_2\times D_8$ x 8, $C_4:D_4$ x 8, $C_2^3:C_4$ x 6, $C_4\times C_8$ x 5, $C_2^2\times D_4$ x 4, $C_2^2\times C_8$ x 3, $C_2^2.D_4$ x 3, $C_2^3:C_4$ x 2, $C_8:C_4$ x 2, $C_2\times C_4^2$

Order 16: $C_2\times D_4$ x 24, $C_2\times C_8$ x 12, $D_8$ x 8, $C_4:C_4$ x 6, $C_2^2:C_4$ x 6, $C_2^2\times C_4$ x 5, $C_4^2$ x 3, $C_2^4$ x 2

Order 8: $C_2\times C_4$ x 21, $D_4$ x 18, $C_2^3$ x 13, $C_8$ x 6

Order 4: $C_2^2$ x 19, $C_4$ x 11

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $(C_4\times C_8).C_4^2$

Order 256: $C_4^2.(C_2^2\times C_4)$, $(C_2\times C_8).D_8$

Order 128: $(C_4\times C_8):C_4$ x 2, $C_2^4.D_4$, $C_4^2.D_4$, $C_4^2.Q_8$, $C_4^2.D_4$, $C_2^3.C_4^2$

Order 64: $C_4:D_8$ x 4, $C_2\times C_4\times C_8$ x 2, $C_4^2:C_4$ x 2, $C_2^2.D_8$, $C_2^4:C_4$, $C_4^2:C_4$, $C_8:C_8$, $C_2^2\times D_8$, $C_4^2:C_2^2$, $C_2^3.D_4$

Order 32: $C_2\times D_8$ x 5, $C_4\times C_8$ x 5, $C_4:D_4$ x 4, $C_2^3:C_4$ x 2, $C_2^2\times D_4$ x 2, $C_2^2\times C_8$ x 2, $C_2^2.D_4$ x 2, $D_4:C_4$, $C_2^3:C_4$, $C_2\times C_4^2$, $C_8:C_4$

Order 16: $C_2\times D_4$ x 10, $C_2\times C_8$ x 6, $C_2^2\times C_4$ x 4, $D_8$ x 3, $C_4^2$ x 3, $C_4:C_4$ x 2, $C_2^2:C_4$ x 2, $C_2^4$

Order 8: $C_2\times C_4$ x 12, $C_2^3$ x 7, $D_4$ x 6, $C_8$ x 3

Order 4: $C_2^2$ x 11, $C_4$ x 6

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $(C_4\times C_8).C_4^2$ ($C_1$)

Order 256: $C_4^2.(C_2^2\times C_4)$ ($C_2$) x 2, $(C_2\times C_8).D_8$ ($C_2$)

Order 128: $(C_4\times C_8):C_4$ ($C_4$) x 4, $C_4^2.Q_8$ ($C_4$), $C_4^2.D_4$ ($C_2^2$), $C_2^3.C_4^2$ ($C_4$)

Order 64: $C_4^2:C_2^2$ ($D_4$) x 2, $C_4:D_8$ ($C_2\times C_4$) x 2, $C_2\times C_4\times C_8$ ($C_2\times C_4$), $C_4:D_8$ ($Q_8$), $C_4:D_8$ ($D_4$)

Order 32: $C_4:D_4$ ($C_4:C_4$) x 2, $C_4:D_4$ ($C_2^2:C_4$) x 2, $C_4\times C_8$ ($C_4:C_4$), $C_4\times C_8$ ($C_4^2$), $C_2\times C_4^2$ ($C_2^2:C_4$)

Order 16: $C_4^2$ ($C_2^3:C_4$), $C_4^2$ ($C_2.C_4^2$), $C_2^2\times C_4$ ($C_2^3:C_4$)

Order 8: $C_2^3$ ($C_4^2:C_4$), $C_2\times C_4$ ($C_4^2:C_4$), $C_2\times C_4$ ($C_2^2.C_4^2$)

Order 4: $C_2^2$ ($(C_4\times C_8):C_4$), $C_2^2$ ($(C_4\times C_8):C_4$), $C_2^2$ ($C_2^4.D_4$)

Order 2: $C_2$ ($C_8:C_8:C_4$) x 2, $C_2$ ($C_4^2.C_4^2$)

Order 1: $C_1$ ($(C_4\times C_8).C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $(C_4\times C_8).C_4^2$ ($C_1$)

Order 256: $C_4^2.(C_2^2\times C_4)$ ($C_2$), $(C_2\times C_8).D_8$ ($C_2$)

Order 128: $C_4^2.Q_8$ ($C_4$), $C_4^2.D_4$ ($C_2^2$), $C_2^3.C_4^2$ ($C_4$), $(C_4\times C_8):C_4$ ($C_4$)

Order 64: $C_2\times C_4\times C_8$ ($C_2\times C_4$), $C_4^2:C_2^2$ ($D_4$), $C_4:D_8$ ($Q_8$), $C_4:D_8$ ($D_4$), $C_4:D_8$ ($C_2\times C_4$)

Order 32: $C_4:D_4$ ($C_4:C_4$), $C_4:D_4$ ($C_2^2:C_4$), $C_4\times C_8$ ($C_4:C_4$), $C_4\times C_8$ ($C_4^2$), $C_2\times C_4^2$ ($C_2^2:C_4$)

Order 16: $C_4^2$ ($C_2^3:C_4$), $C_4^2$ ($C_2.C_4^2$), $C_2^2\times C_4$ ($C_2^3:C_4$)

Order 8: $C_2^3$ ($C_4^2:C_4$), $C_2\times C_4$ ($C_4^2:C_4$), $C_2\times C_4$ ($C_2^2.C_4^2$)

Order 4: $C_2^2$ ($(C_4\times C_8):C_4$), $C_2^2$ ($(C_4\times C_8):C_4$), $C_2^2$ ($C_2^4.D_4$)

Order 2: $C_2$ ($C_8:C_8:C_4$), $C_2$ ($C_4^2.C_4^2$)

Order 1: $C_1$ ($(C_4\times C_8).C_4^2$)

Series

Derived series

$(C_4\times C_8).C_4^2$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_1$

Chief series

$(C_4\times C_8).C_4^2$

$\rhd$

$(C_2\times C_8).D_8$

$\rhd$

$C_4^2.D_4$

$\rhd$

$C_2\times C_4\times C_8$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_4\times C_8).C_4^2$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2\times C_4^2$

$\lhd$

$C_4^2.D_4$

$\lhd$

$(C_4\times C_8).C_4^2$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $38 \times 38$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $27 \times 27$ rational character table.