Abstract group 256.12589: $C_8^2:C_2^2$

• Label: 256.12589
• Order: \( 2^{8} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{20} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{15} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $128$
• Rank: $4$

Group information

Description:	$C_8^2:C_2^2$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$(C_2^5\times C_4^2).C_2^6.A_4.D_4$, of order \(3145728\)\(\medspace = 2^{20} \cdot 3 \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$3$
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	135	24	96	256
Conjugacy classes	1	15	12	48	76
Divisions	1	15	12	24	52
Autjugacy classes	1	3	2	1	7

Dimension	1	2	4
Irr. complex chars.	16	60	0	76
Irr. rational chars.	16	12	24	52

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$128$
Rank:	$4$
Inequivalent generating quadruples:	$420$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{8}=d^{8}=[a,b]=[a,c]=[a,d]=[c,d]=1, c^{b}=c^{7}, d^{b}=d^{7} \rangle$
Permutation group:	Degree $18$ $\langle(2,5)(3,6)(7,8)(9,10)(12,15)(13,16)(17,18), (1,2,4,5)(3,6,7,8)(11,12,16,18,14,17,13,15) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 0 & 15 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 16 \\ 16 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 17 & 19 \\ 0 & 15 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
	$\left\langle \left(\begin{array}{rr} 39 & 40 \\ 32 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 35 & 30 \\ 32 & 21 \end{array}\right), \left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 27 & 7 \\ 0 & 27 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 41 & 0 \\ 0 & 41 \end{array}\right), \left(\begin{array}{rr} 41 & 8 \\ 40 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/56\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_8:D_8)$
Semidirect product:	$C_8^2$ $\,\rtimes\,$ $C_2^2$	$(C_2\times C_8)$ $\,\rtimes\,$ $D_8$	$C_8$ $\,\rtimes\,$ $(C_2\times D_8)$	$(C_4^2.D_4)$ $\,\rtimes\,$ $C_2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^2$ . $C_2^4$	$(C_4\times C_8)$ . $D_4$	$C_4$ . $(C_4:D_8)$	$(C_2^2\times C_8)$ . $D_4$	all 20

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{32}\Z)$.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5} \times C_{8}$
Commutator length:	$1$

Subgroups

There are 2447 subgroups in 671 conjugacy classes, 191 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_4:D_4$
Commutator:	$G' \simeq$ $C_4^2$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_4^2$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_8^2:C_2^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_8^2:C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_4:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8^2:C_2^2$

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 256: $C_8^2:C_2^2$

Order 128: $C_8:D_8$ x 8, $C_4^2.D_4$ x 6, $C_2\times C_8^2$

Order 64: $C_4:D_8$ x 36, $C_2^2\times D_8$ x 12, $C_4^2:C_2^2$ x 4, $C_8^2$ x 4, $C_2\times C_4\times C_8$ x 3

Order 32: $C_2\times D_8$ x 96, $C_4:D_4$ x 20, $C_2^2\times D_4$ x 12, $C_4\times C_8$ x 12, $C_2^2\times C_8$ x 6, $C_2\times C_4^2$

Order 16: $D_8$ x 96, $C_2\times D_4$ x 84, $C_2\times C_8$ x 36, $C_4^2$ x 4, $C_2^4$ x 4, $C_2^2\times C_4$ x 3

Order 8: $D_4$ x 72, $C_2^3$ x 33, $C_8$ x 24, $C_2\times C_4$ x 18

Order 4: $C_2^2$ x 47, $C_4$ x 12

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_8^2:C_2^2$

Order 128: $C_8:D_8$, $C_4^2.D_4$, $C_2\times C_8^2$

Order 64: $C_4:D_8$ x 2, $C_2\times C_4\times C_8$, $C_2^2\times D_8$, $C_4^2:C_2^2$, $C_8^2$

Order 32: $C_2\times D_8$ x 2, $C_4:D_4$ x 2, $C_4\times C_8$ x 2, $C_2^2\times D_4$, $C_2^2\times C_8$, $C_2\times C_4^2$

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

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_8^2:C_2^2$ ($C_1$)

Order 128: $C_8:D_8$ ($C_2$) x 8, $C_4^2.D_4$ ($C_2$) x 6, $C_2\times C_8^2$ ($C_2$)

Order 64: $C_4:D_8$ ($C_2^2$) x 24, $C_4^2:C_2^2$ ($C_2^2$) x 4, $C_8^2$ ($C_2^2$) x 4, $C_2\times C_4\times C_8$ ($C_2^2$) x 3

Order 32: $C_4:D_4$ ($C_2^3$) x 8, $C_2^2\times C_8$ ($D_4$) x 6, $C_4\times C_8$ ($C_2^3$) x 6, $C_4\times C_8$ ($D_4$) x 6, $C_2\times C_4^2$ ($C_2^3$)

Order 16: $C_2\times C_8$ ($D_8$) x 24, $C_2\times C_8$ ($C_2\times D_4$) x 12, $C_4^2$ ($C_2\times D_4$) x 3, $C_2^2\times C_4$ ($C_2\times D_4$) x 3, $C_4^2$ ($C_2^4$)

Order 8: $C_8$ ($C_2\times D_8$) x 24, $C_2\times C_4$ ($C_2\times D_8$) x 12, $C_2\times C_4$ ($C_2^2\times D_4$) x 3, $C_2\times C_4$ ($C_4:D_4$) x 3, $C_2^3$ ($C_4:D_4$)

Order 4: $C_2^2$ ($C_4:D_8$) x 6, $C_4$ ($C_2^2\times D_8$) x 6, $C_4$ ($C_4:D_8$) x 6, $C_2^2$ ($C_4^2:C_2^2$)

Order 2: $C_2$ ($C_8:D_8$) x 4, $C_2$ ($C_4^2.D_4$) x 3

Order 1: $C_1$ ($C_8^2:C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_8^2:C_2^2$ ($C_1$)

Order 128: $C_8:D_8$ ($C_2$), $C_4^2.D_4$ ($C_2$), $C_2\times C_8^2$ ($C_2$)

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

Order 32: $C_2^2\times C_8$ ($D_4$), $C_4:D_4$ ($C_2^3$), $C_4\times C_8$ ($C_2^3$), $C_4\times C_8$ ($D_4$), $C_2\times C_4^2$ ($C_2^3$)

Order 16: $C_2\times C_8$ ($D_8$), $C_2\times C_8$ ($C_2\times D_4$), $C_4^2$ ($C_2^4$), $C_4^2$ ($C_2\times D_4$), $C_2^2\times C_4$ ($C_2\times D_4$)

Order 8: $C_2^3$ ($C_4:D_4$), $C_2\times C_4$ ($C_2^2\times D_4$), $C_2\times C_4$ ($C_2\times D_8$), $C_2\times C_4$ ($C_4:D_4$), $C_8$ ($C_2\times D_8$)

Order 4: $C_2^2$ ($C_4^2:C_2^2$), $C_2^2$ ($C_4:D_8$), $C_4$ ($C_2^2\times D_8$), $C_4$ ($C_4:D_8$)

Order 2: $C_2$ ($C_8:D_8$), $C_2$ ($C_4^2.D_4$)

Order 1: $C_1$ ($C_8^2:C_2^2$)

Series

Derived series

$C_8^2:C_2^2$

$\rhd$

$C_4^2$

$\rhd$

$C_1$

Chief series

$C_8^2:C_2^2$

$\rhd$

$C_8:D_8$

$\rhd$

$C_8^2$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_8^2:C_2^2$

$\rhd$

$C_4^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2\times C_4^2$

$\lhd$

$C_8^2:C_2^2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

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

Character theory

Complex character table

See the $76 \times 76$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $52 \times 52$ rational character table.