Abstract group 512.2030: $C_4.D_{64}$

• Label: 512.2030
• Order: \( 2^{9} \)
• Exponent: \( 2^{7} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{13} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $256$
• Trans deg.: $256$
• Rank: $2$

Group information

Description:	$C_4.D_{64}$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(128\)\(\medspace = 2^{7} \)
Automorphism group:	$C_{32}.C_{32}.C_2^3$, of order \(8192\)\(\medspace = 2^{13} \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$7$
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	32	64	128
Elements	1	67	68	136	16	32	64	128	512
Conjugacy classes	1	3	4	6	8	16	32	64	134
Divisions	1	3	3	3	2	2	2	1	17
Autjugacy classes	1	3	3	3	2	2	2	1	17

Dimension	1	2	4	8	16	32	128
Irr. complex chars.	8	126	0	0	0	0	0	134
Irr. rational chars.	4	4	2	2	2	2	1	17

Minimal presentations

Permutation degree:	$256$
Transitive degree:	$256$
Rank:	$2$
Inequivalent generating pairs:	$12$

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:	$\langle a, b, c \mid b^{128}=c^{2}=[b,c]=1, a^{2}=b^{32}c, b^{a}=b^{95}c, c^{a}=b^{64}c \rangle$
Permutation group:	Degree $256$ $\langle(1,131,3,130,2,132,4,129)(5,135,7,134,6,136,8,133)(9,139,11,138,10,140,12,137) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 141 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 81 & 0 \\ 0 & 165 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{257})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{64}.C_4)$ $\,\rtimes\,$ $C_2$	$(C_2\times C_{128})$ $\,\rtimes\,$ $C_2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{64}$ . $C_4$	$C_{64}$ . $D_4$	$C_4$ . $D_{64}$	$Q_{128}$ . $C_4$	all 20

Elements of the group are displayed as matrices in $\GL_{2}(\F_{257})$.

Homology

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

Subgroups

There are 402 subgroups in 52 conjugacy classes, 24 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $D_{64}$
Commutator:	$G' \simeq$ $C_{64}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_{64}$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_4.D_{64}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_4.D_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $D_{32}:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_{64}$

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: $C_4.D_{64}$

Order 256: $D_{64}:C_2$, $C_2\times C_{128}$, $C_{64}.C_4$

Order 128: $D_{32}:C_2$, $Q_{128}$, $\SD_{128}$, $D_{64}$, $C_2\times C_{64}$, $C_{32}.C_4$, $C_{128}$

Order 64: $C_{64}$ x 2, $Q_{64}$, $\SD_{64}$, $D_{32}$, $C_2\times C_{32}$, $C_{16}.C_4$, $D_{16}:C_2$

Order 32: $C_{32}$ x 2, $D_8:C_2$, $Q_{32}$, $\SD_{32}$, $D_{16}$, $C_2\times C_{16}$, $C_8.C_4$

Order 16: $C_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_2\times C_8$, $D_4:C_2$

Order 8: $C_8$ x 3, $D_4$ x 2, $C_2\times C_4$ x 2, $Q_8$

Order 4: $C_4$ x 3, $C_2^2$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $C_4.D_{64}$

Order 256: $D_{64}:C_2$, $C_2\times C_{128}$, $C_{64}.C_4$

Order 128: $D_{32}:C_2$, $Q_{128}$, $\SD_{128}$, $D_{64}$, $C_2\times C_{64}$, $C_{32}.C_4$, $C_{128}$

Order 64: $C_{64}$ x 2, $Q_{64}$, $\SD_{64}$, $D_{32}$, $C_2\times C_{32}$, $C_{16}.C_4$, $D_{16}:C_2$

Order 32: $C_{32}$ x 2, $D_8:C_2$, $Q_{32}$, $\SD_{32}$, $D_{16}$, $C_2\times C_{16}$, $C_8.C_4$

Order 16: $C_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_2\times C_8$, $D_4:C_2$

Order 8: $C_8$ x 3, $D_4$ x 2, $C_2\times C_4$ x 2, $Q_8$

Order 4: $C_4$ x 3, $C_2^2$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $C_4.D_{64}$ ($C_1$)

Order 256: $D_{64}:C_2$ ($C_2$), $C_2\times C_{128}$ ($C_2$), $C_{64}.C_4$ ($C_2$)

Order 128: $Q_{128}$ ($C_4$), $D_{64}$ ($C_4$), $C_2\times C_{64}$ ($C_2^2$)

Order 64: $C_2\times C_{32}$ ($D_4$), $C_{64}$ ($D_4$), $C_{64}$ ($C_2\times C_4$)

Order 32: $C_2\times C_{16}$ ($D_8$), $C_{32}$ ($\SD_{16}$), $C_{32}$ ($C_2^2:C_4$)

Order 16: $C_2\times C_8$ ($D_{16}$), $C_{16}$ ($D_4:C_4$), $C_{16}$ ($\SD_{32}$)

Order 8: $C_2\times C_4$ ($D_{32}$), $C_8$ ($\SD_{64}$), $C_8$ ($D_8:C_4$)

Order 4: $C_2^2$ ($\SD_{128}$), $C_4$ ($D_{64}$), $C_4$ ($D_{16}:C_4$)

Order 2: $C_2$ ($D_{32}:C_4$)

Order 1: $C_1$ ($C_4.D_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $C_4.D_{64}$ ($C_1$)

Order 256: $D_{64}:C_2$ ($C_2$), $C_2\times C_{128}$ ($C_2$), $C_{64}.C_4$ ($C_2$)

Order 128: $Q_{128}$ ($C_4$), $D_{64}$ ($C_4$), $C_2\times C_{64}$ ($C_2^2$)

Order 64: $C_2\times C_{32}$ ($D_4$), $C_{64}$ ($D_4$), $C_{64}$ ($C_2\times C_4$)

Order 32: $C_2\times C_{16}$ ($D_8$), $C_{32}$ ($\SD_{16}$), $C_{32}$ ($C_2^2:C_4$)

Order 16: $C_2\times C_8$ ($D_{16}$), $C_{16}$ ($D_4:C_4$), $C_{16}$ ($\SD_{32}$)

Order 8: $C_2\times C_4$ ($D_{32}$), $C_8$ ($\SD_{64}$), $C_8$ ($D_8:C_4$)

Order 4: $C_2^2$ ($\SD_{128}$), $C_4$ ($D_{64}$), $C_4$ ($D_{16}:C_4$)

Order 2: $C_2$ ($D_{32}:C_4$)

Order 1: $C_1$ ($C_4.D_{64}$)

Series

Derived series

$C_4.D_{64}$

$\rhd$

$C_{64}$

$\rhd$

$C_1$

Chief series

$C_4.D_{64}$

$\rhd$

$C_2\times C_{128}$

$\rhd$

$C_2\times C_{64}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4.D_{64}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_4$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_2\times C_{16}$

$\lhd$

$C_2\times C_{32}$

$\lhd$

$C_2\times C_{64}$

$\lhd$

$C_4.D_{64}$

Supergroups

This group is a maximal subgroup of 1 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 $134 \times 134$ 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 $17 \times 17$ rational character table.