Abstract group 512.2043: $\SD_{512}$

• Label: 512.2043
• Order: \( 2^{9} \)
• Exponent: \( 2^{8} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{14} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $256$
• Trans deg.: $256$
• Rank: $2$

Group information

Description:	$\SD_{512}$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(256\)\(\medspace = 2^{8} \)
Automorphism group:	$C_{64}.C_{64}.C_2^2$, of order \(16384\)\(\medspace = 2^{14} \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$8$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16	32	64	128	256
Elements	1	129	130	4	8	16	32	64	128	512
Conjugacy classes	1	2	2	2	4	8	16	32	64	131
Divisions	1	2	2	1	1	1	1	1	1	11
Autjugacy classes	1	2	2	1	1	1	1	1	1	11

Dimension	1	2	4	8	16	32	64	128
Irr. complex chars.	4	127	0	0	0	0	0	0	131
Irr. rational chars.	4	1	1	1	1	1	1	1	11

Minimal presentations

Permutation degree:	$256$
Transitive degree:	$256$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	128
Arbitrary	2	4	128

Constructions

Presentation:	$\langle a, b \mid b^{256}=1, a^{2}=b^{128}, b^{a}=b^{127} \rangle$
Permutation group:	Degree $256$ $\langle(1,130,2,129)(3,132,4,131)(5,134,6,133)(7,136,8,135)(9,138,10,137)(11,140,12,139) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 126 \end{array}\right), \left(\begin{array}{rr} 35 & 73 \\ 109 & 35 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{127})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$Q_{256}$ $\,\rtimes\,$ $C_2$	$C_{256}$ $\,\rtimes\,$ $C_2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{64}$ . $D_4$	$C_{32}$ . $D_8$	$C_8$ . $D_{32}$	$C_4$ . $D_{64}$	all 8

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 392 subgroups in 25 conjugacy classes, 12 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{128}$
Commutator:	$G' \simeq$ $C_{128}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_{128}$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $\SD_{512}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $\SD_{512}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $D_{128}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{512}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 512: $\SD_{512}$

Order 256: $Q_{256}$, $D_{128}$, $C_{256}$

Order 128: $Q_{128}$, $D_{64}$, $C_{128}$

Order 64: $Q_{64}$, $D_{32}$, $C_{64}$

Order 32: $Q_{32}$, $D_{16}$, $C_{32}$

Order 16: $Q_{16}$, $D_8$, $C_{16}$

Order 8: $Q_8$, $D_4$, $C_8$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $\SD_{512}$

Order 256: $Q_{256}$, $D_{128}$, $C_{256}$

Order 128: $Q_{128}$, $D_{64}$, $C_{128}$

Order 64: $Q_{64}$, $D_{32}$, $C_{64}$

Order 32: $Q_{32}$, $D_{16}$, $C_{32}$

Order 16: $Q_{16}$, $D_8$, $C_{16}$

Order 8: $Q_8$, $D_4$, $C_8$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $\SD_{512}$ ($C_1$)

Order 256: $Q_{256}$ ($C_2$), $D_{128}$ ($C_2$), $C_{256}$ ($C_2$)

Order 128: $C_{128}$ ($C_2^2$)

Order 64: $C_{64}$ ($D_4$)

Order 32: $C_{32}$ ($D_8$)

Order 16: $C_{16}$ ($D_{16}$)

Order 8: $C_8$ ($D_{32}$)

Order 4: $C_4$ ($D_{64}$)

Order 2: $C_2$ ($D_{128}$)

Order 1: $C_1$ ($\SD_{512}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $\SD_{512}$ ($C_1$)

Order 256: $Q_{256}$ ($C_2$), $D_{128}$ ($C_2$), $C_{256}$ ($C_2$)

Order 128: $C_{128}$ ($C_2^2$)

Order 64: $C_{64}$ ($D_4$)

Order 32: $C_{32}$ ($D_8$)

Order 16: $C_{16}$ ($D_{16}$)

Order 8: $C_8$ ($D_{32}$)

Order 4: $C_4$ ($D_{64}$)

Order 2: $C_2$ ($D_{128}$)

Order 1: $C_1$ ($\SD_{512}$)

Series

Derived series

$\SD_{512}$

$\rhd$

$C_{128}$

$\rhd$

$C_1$

Chief series

$\SD_{512}$

$\rhd$

$C_{256}$

$\rhd$

$C_{128}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$\SD_{512}$

$\rhd$

$C_{128}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$C_8$

$\lhd$

$C_{16}$

$\lhd$

$C_{32}$

$\lhd$

$C_{64}$

$\lhd$

$C_{128}$

$\lhd$

$\SD_{512}$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $131 \times 131$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	4A	4B	8A	16A	32A	64A	128A	256A
	Size	1	1	128	2	128	4	8	16	32	64	128
	2 P	1A	1A	1A	2A	2A	4A	8A	16A	32A	64A	128A
512.2043.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
512.2043.1b		$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
512.2043.1c		$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$
512.2043.1d		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$
512.2043.2a		$2$	$2$	$0$	$2$	$0$	$2$	$2$	$2$	$2$	$-2$	$0$
512.2043.2b		$4$	$4$	$0$	$4$	$0$	$4$	$4$	$4$	$-4$	$0$	$0$
512.2043.2c		$8$	$8$	$0$	$8$	$0$	$8$	$8$	$-8$	$0$	$0$	$0$
512.2043.2d		$16$	$16$	$0$	$16$	$0$	$16$	$-16$	$0$	$0$	$0$	$0$
512.2043.2e		$32$	$32$	$0$	$32$	$0$	$-32$	$0$	$0$	$0$	$0$	$0$
512.2043.2f		$64$	$64$	$0$	$-64$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
512.2043.2g		$128$	$-128$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$