Abstract group 256.53038: $C_2^3\times C_4\times C_8$

• Label: 256.53038
• Order: \( 2^{8} \)
• Exponent: \( 2^{3} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{22} \cdot 3 \cdot 7 \)
• Perm deg.: $18$
• Trans deg.: $256$
• Rank: $5$

Group information

Description:	$C_2^3\times C_4\times C_8$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^6.C_2^5.C_2^6.C_2^2.\PSL(2,7)$, of order \(88080384\)\(\medspace = 2^{22} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 8
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
Elements	1	31	96	128	256
Conjugacy classes	1	31	96	128	256
Divisions	1	31	48	32	112
Autjugacy classes	1	3	3	1	8

Dimension	1	2	4
Irr. complex chars.	256	0	0	256
Irr. rational chars.	32	48	32	112

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$256$
Rank:	$5$
Inequivalent generating 5-tuples:	$3720$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{2}=d^{4}=e^{8}=1 \rangle$
Permutation group:	Degree $18$ $\langle(11,18,14,16,12,17,13,15), (7,10,8,9), (5,6), (1,2), (3,4), (11,14,12,13)(15,18,16,17), (7,8)(9,10), (11,12)(13,14)(15,16)(17,18)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 15 & 2 \\ 4 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 8 \\ 8 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 12 \\ 12 & 5 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 12 & 11 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 12 & 13 \end{array}\right), \left(\begin{array}{rr} 13 & 20 \\ 20 & 9 \end{array}\right), \left(\begin{array}{rr} 17 & 12 \\ 12 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $C_4$ $\, \times\, $ $C_8$
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:	$C_2^4$ . $C_4^2$	$C_4^2$ . $C_2^4$	$(C_2^4\times C_8)$ . $C_2$	$(C_2^4\times C_4)$ . $C_4$	all 35

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

Homology

Primary decomposition:	$C_{2}^{3} \times C_{4} \times C_{8}$
Schur multiplier:	$C_{2}^{9} \times C_{4}$
Commutator length:	$0$

Subgroups

There are 2319 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:	$Z \simeq$ $C_2^3\times C_4\times C_8$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2^3\times C_4\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2^5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_4\times C_8$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^3\times C_4\times C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times C_4\times C_8$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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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: $C_2^3\times C_4\times C_8$

Order 128: $C_2^2\times C_4\times C_8$ x 28, $C_2^4\times C_8$ x 2, $C_2^3\times C_4^2$

Order 64: $C_2\times C_4\times C_8$ x 112, $C_2^3\times C_8$ x 60, $C_2^2\times C_4^2$ x 28, $C_2^4\times C_4$ x 3

Order 32: $C_2^2\times C_8$ x 280, $C_2\times C_4^2$ x 112, $C_2^3\times C_4$ x 90, $C_4\times C_8$ x 64, $C_2^5$

Order 16: $C_2^2\times C_4$ x 420, $C_2\times C_8$ x 240, $C_4^2$ x 64, $C_2^4$ x 31

Order 8: $C_2\times C_4$ x 360, $C_2^3$ x 155, $C_8$ x 32

Order 4: $C_2^2$ x 155, $C_4$ x 48

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_2^3\times C_4\times C_8$

Order 128: $C_2^4\times C_8$, $C_2^3\times C_4^2$, $C_2^2\times C_4\times C_8$

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

Order 32: $C_2^3\times C_4$ x 5, $C_2^2\times C_8$ x 2, $C_2\times C_4^2$ x 2, $C_2^5$, $C_4\times C_8$

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

Order 8: $C_2\times C_4$ x 6, $C_2^3$ x 4, $C_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_2^3\times C_4\times C_8$ ($C_1$)

Order 128: $C_2^2\times C_4\times C_8$ ($C_2$) x 28, $C_2^4\times C_8$ ($C_2$) x 2, $C_2^3\times C_4^2$ ($C_2$)

Order 64: $C_2\times C_4\times C_8$ ($C_2^2$) x 112, $C_2^3\times C_8$ ($C_4$) x 32, $C_2^3\times C_8$ ($C_2^2$) x 28, $C_2^2\times C_4^2$ ($C_2^2$) x 14, $C_2^2\times C_4^2$ ($C_4$) x 14, $C_2^4\times C_4$ ($C_4$) x 2, $C_2^4\times C_4$ ($C_2^2$)

Order 32: $C_2^2\times C_8$ ($C_2\times C_4$) x 224, $C_2\times C_4^2$ ($C_2\times C_4$) x 84, $C_4\times C_8$ ($C_2^3$) x 64, $C_2^2\times C_8$ ($C_2^3$) x 56, $C_2^3\times C_4$ ($C_2\times C_4$) x 51, $C_2^3\times C_4$ ($C_8$) x 32, $C_2\times C_4^2$ ($C_2^3$) x 28, $C_2^3\times C_4$ ($C_2^3$) x 7, $C_2^5$ ($C_2\times C_4$)

Order 16: $C_2\times C_8$ ($C_2^2\times C_4$) x 224, $C_2^2\times C_4$ ($C_2\times C_8$) x 224, $C_2^2\times C_4$ ($C_2^2\times C_4$) x 133, $C_4^2$ ($C_2^2\times C_4$) x 56, $C_2^2\times C_4$ ($C_4^2$) x 56, $C_2\times C_8$ ($C_2^4$) x 16, $C_2^4$ ($C_2\times C_8$) x 16, $C_4^2$ ($C_2^4$) x 8, $C_2^4$ ($C_4^2$) x 8, $C_2^4$ ($C_2^2\times C_4$) x 7, $C_2^2\times C_4$ ($C_2^4$) x 7

Order 8: $C_2\times C_4$ ($C_2^2\times C_8$) x 224, $C_2\times C_4$ ($C_2\times C_4^2$) x 84, $C_2^3$ ($C_4\times C_8$) x 64, $C_2^3$ ($C_2^2\times C_8$) x 56, $C_2\times C_4$ ($C_2^3\times C_4$) x 51, $C_8$ ($C_2^3\times C_4$) x 32, $C_2^3$ ($C_2\times C_4^2$) x 28, $C_2^3$ ($C_2^3\times C_4$) x 7, $C_2\times C_4$ ($C_2^5$)

Order 4: $C_2^2$ ($C_2\times C_4\times C_8$) x 112, $C_4$ ($C_2^3\times C_8$) x 32, $C_2^2$ ($C_2^3\times C_8$) x 28, $C_2^2$ ($C_2^2\times C_4^2$) x 14, $C_4$ ($C_2^2\times C_4^2$) x 14, $C_4$ ($C_2^4\times C_4$) x 2, $C_2^2$ ($C_2^4\times C_4$)

Order 2: $C_2$ ($C_2^2\times C_4\times C_8$) x 28, $C_2$ ($C_2^4\times C_8$) x 2, $C_2$ ($C_2^3\times C_4^2$)

Order 1: $C_1$ ($C_2^3\times C_4\times C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_2^3\times C_4\times C_8$ ($C_1$)

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

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

Order 32: $C_2^3\times C_4$ ($C_2\times C_4$) x 3, $C_2^5$ ($C_2\times C_4$), $C_2^3\times C_4$ ($C_2^3$), $C_2^3\times C_4$ ($C_8$), $C_2^2\times C_8$ ($C_2^3$), $C_2^2\times C_8$ ($C_2\times C_4$), $C_4\times C_8$ ($C_2^3$), $C_2\times C_4^2$ ($C_2^3$), $C_2\times C_4^2$ ($C_2\times C_4$)

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

Order 8: $C_2\times C_4$ ($C_2^3\times C_4$) x 3, $C_2^3$ ($C_2^3\times C_4$), $C_2^3$ ($C_2^2\times C_8$), $C_2^3$ ($C_4\times C_8$), $C_2^3$ ($C_2\times C_4^2$), $C_2\times C_4$ ($C_2^5$), $C_2\times C_4$ ($C_2^2\times C_8$), $C_2\times C_4$ ($C_2\times C_4^2$), $C_8$ ($C_2^3\times C_4$)

Order 4: $C_2^2$ ($C_2\times C_4\times C_8$), $C_2^2$ ($C_2^4\times C_4$), $C_2^2$ ($C_2^3\times C_8$), $C_2^2$ ($C_2^2\times C_4^2$), $C_4$ ($C_2^4\times C_4$), $C_4$ ($C_2^3\times C_8$), $C_4$ ($C_2^2\times C_4^2$)

Order 2: $C_2$ ($C_2^4\times C_8$), $C_2$ ($C_2^3\times C_4^2$), $C_2$ ($C_2^2\times C_4\times C_8$)

Order 1: $C_1$ ($C_2^3\times C_4\times C_8$)

Series

Derived series

$C_2^3\times C_4\times C_8$

$\rhd$

$C_1$

Chief series

$C_2^3\times C_4\times C_8$

$\rhd$

$C_2^2\times C_4\times C_8$

$\rhd$

$C_2\times C_4\times C_8$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3\times C_4\times C_8$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3\times C_4\times C_8$

Supergroups

This group is a maximal subgroup of 57 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $256 \times 256$ 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 $112 \times 112$ rational character table.