Abstract group $C_2\times C_{16}^2$

• Label: 512.29399
• Order: \( 2^{9} \)
• Exponent: \( 2^{4} \)
• Abelian: yes
• $\card{\operatorname{Aut}(G)}$: \( 2^{17} \cdot 3 \)
• Perm deg.: $34$
• Trans deg.: $512$
• Rank: $3$

Group information

Description:	$C_2\times C_{16}^2$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	Group of order \(393216\)\(\medspace = 2^{17} \cdot 3 \) (generators)
Outer automorphisms:	Group of order \(393216\)\(\medspace = 2^{17} \cdot 3 \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	4	8	16
Elements	1	7	24	96	384	512
Conjugacy classes	1	7	24	96	384	512
Divisions	1	7	12	24	48	92
Autjugacy classes	1	2	2	2	1	8

Dimension	1	2	4	8
Irr. complex chars.	512	0	0	0	512
Irr. rational chars.	8	12	24	48	92

Minimal Presentations

Permutation degree:	$34$
Transitive degree:	$512$
Rank:	$3$
Inequivalent generating triples:	$112$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	3	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b, c \mid a^{2}=b^{16}=c^{16}=1 \rangle$
Permutation group:	Degree $34$ $\langle(3,18,10,14,6,16,8,12,4,17,9,13,5,15,7,11), (19,34,26,30,22,32,24,28,20,33,25,29,21,31,23,27) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $C_{16}$ ${}^2$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_8^2$ . $C_2^3$	$C_2^3$ . $C_8^2$	$(C_8\times C_{16})$ . $C_4$	$(C_4\times C_{16})$ . $C_8$	all 34
Aut. group:	$\Aut(C_{1088})$

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

Homology

Primary decomposition:	$C_{2} \times C_{16}^{2}$
Schur multiplier:	$C_{2}^{2} \times C_{16}$
Commutator length:	$0$

Subgroups

There are 322 subgroups, all normal (8 characteristic).

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

Special subgroups

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

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

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

Order 256: $C_{16}^2$ x 4, $C_2\times C_8\times C_{16}$ x 3

Order 128: $C_8\times C_{16}$ x 12, $C_2\times C_4\times C_{16}$ x 6, $C_2\times C_8^2$

Order 64: $C_4\times C_{16}$ x 24, $C_2^2\times C_{16}$ x 12, $C_8^2$ x 4, $C_2\times C_4\times C_8$ x 3

Order 32: $C_2\times C_{16}$ x 72, $C_4\times C_8$ x 12, $C_2^2\times C_8$ x 6, $C_2\times C_4^2$

Order 16: $C_{16}$ x 48, $C_2\times C_8$ x 36, $C_4^2$ x 4, $C_2^2\times C_4$ x 3

Order 8: $C_8$ x 24, $C_2\times C_4$ x 18, $C_2^3$

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 256: $C_2\times C_8\times C_{16}$, $C_{16}^2$

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

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

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

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

Order 8: $C_2\times C_4$ x 3, $C_8$ x 2, $C_2^3$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

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

Order 128: $C_2\times C_4\times C_{16}$ ($C_4$) x 6, $C_8\times C_{16}$ ($C_2^2$) x 6, $C_8\times C_{16}$ ($C_4$) x 6, $C_2\times C_8^2$ ($C_2^2$)

Order 64: $C_4\times C_{16}$ ($C_2\times C_4$) x 12, $C_4\times C_{16}$ ($C_8$) x 12, $C_2^2\times C_{16}$ ($C_8$) x 12, $C_2\times C_4\times C_8$ ($C_2\times C_4$) x 3, $C_8^2$ ($C_2\times C_4$) x 3, $C_8^2$ ($C_2^3$)

Order 32: $C_2\times C_{16}$ ($C_{16}$) x 48, $C_2\times C_{16}$ ($C_2\times C_8$) x 24, $C_2^2\times C_8$ ($C_2\times C_8$) x 6, $C_4\times C_8$ ($C_2\times C_8$) x 6, $C_4\times C_8$ ($C_4^2$) x 3, $C_4\times C_8$ ($C_2^2\times C_4$) x 3, $C_2\times C_4^2$ ($C_4^2$)

Order 16: $C_{16}$ ($C_2\times C_{16}$) x 48, $C_2\times C_8$ ($C_2\times C_{16}$) x 24, $C_2\times C_8$ ($C_2^2\times C_8$) x 6, $C_2\times C_8$ ($C_4\times C_8$) x 6, $C_4^2$ ($C_4\times C_8$) x 3, $C_2^2\times C_4$ ($C_4\times C_8$) x 3, $C_4^2$ ($C_2\times C_4^2$)

Order 8: $C_2\times C_4$ ($C_4\times C_{16}$) x 12, $C_8$ ($C_4\times C_{16}$) x 12, $C_8$ ($C_2^2\times C_{16}$) x 12, $C_2\times C_4$ ($C_2\times C_4\times C_8$) x 3, $C_2\times C_4$ ($C_8^2$) x 3, $C_2^3$ ($C_8^2$)

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

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

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

Normal subgroups up to automorphism (quotient in parentheses)

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

Order 256: $C_2\times C_8\times C_{16}$ ($C_2$), $C_{16}^2$ ($C_2$)

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

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

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

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

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

Order 4: $C_2^2$ ($C_8\times C_{16}$), $C_2^2$ ($C_2\times C_8^2$), $C_4$ ($C_2\times C_4\times C_{16}$), $C_4$ ($C_8\times C_{16}$)

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

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

Series

Derived series

$C_2\times C_{16}^2$

$\rhd$

$C_1$

Chief series

$C_2\times C_{16}^2$

$\rhd$

$C_{16}^2$

$\rhd$

$C_8\times C_{16}$

$\rhd$

$C_8^2$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2\times C_{16}^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{16}^2$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

The $512 \times 512$ character table is not available for this group.

Rational character table

See the $92 \times 92$ rational character table.