Abstract group 2048.clc: $C_8\times C_{256}$

• Label: 2048.clc
• Order: \( 2^{11} \)
• Exponent: \( 2^{8} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{15} \)
• Perm deg.: $264$
• Trans deg.: $2048$
• Rank: $2$

Group information

Description:	$C_8\times C_{256}$
Order:	\(2048\)\(\medspace = 2^{11} \)
Exponent:	\(256\)\(\medspace = 2^{8} \)
Automorphism group:	$C_2.C_4^3.C_4.C_2^6$, of order \(32768\)\(\medspace = 2^{15} \)
Composition factors:	$C_2$ x 11
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	4	8	16	32	64	128	256
Elements	1	3	12	48	64	128	256	512	1024	2048
Conjugacy classes	1	3	12	48	64	128	256	512	1024	2048
Divisions	1	3	6	12	8	8	8	8	8	62
Autjugacy classes	1	2	3	4	4	4	3	2	1	24

Minimal presentations

Permutation degree:	$264$
Transitive degree:	$2048$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	2	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{8}=b^{256}=1 \rangle$
Permutation group:	Degree $264$ $\langle(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 222 & 0 \\ 0 & 66 \end{array}\right), \left(\begin{array}{rr} 64 & 0 \\ 0 & 253 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{257})$
Direct product:	$C_8$ $\, \times\, $ $C_{256}$
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:	$C_{64}$ . $C_{32}$ (4)	$C_{32}$ . $C_{64}$ (4)	$C_{128}$ . $C_{16}$ (4)	$C_8^2$ . $C_{32}$	all 56

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

Homology

Primary decomposition:	$C_{8} \times C_{256}$
Schur multiplier:	$C_{8}$
Commutator length:	$0$

Subgroups

There are 112 subgroups, all normal (36 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_8\times C_{256}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_8\times C_{256}$
Frattini:	$\Phi \simeq$ $C_4\times C_{128}$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_8\times C_{256}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_8\times C_{256}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_4\times C_{128}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times C_{256}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 2048: $C_8\times C_{256}$

Order 1024: $C_4\times C_{256}$ x 2, $C_8\times C_{128}$

Order 512: $C_2\times C_{256}$ x 4, $C_4\times C_{128}$ x 2, $C_8\times C_{64}$

Order 256: $C_{256}$ x 8, $C_2\times C_{128}$ x 4, $C_4\times C_{64}$ x 2, $C_8\times C_{32}$

Order 128: $C_{128}$ x 8, $C_2\times C_{64}$ x 4, $C_4\times C_{32}$ x 2, $C_8\times C_{16}$

Order 64: $C_{64}$ x 8, $C_2\times C_{32}$ x 4, $C_4\times C_{16}$ x 2, $C_8^2$

Order 32: $C_{32}$ x 8, $C_2\times C_{16}$ x 4, $C_4\times C_8$ x 3

Order 16: $C_{16}$ x 8, $C_2\times C_8$ x 6, $C_4^2$

Order 8: $C_8$ x 12, $C_2\times C_4$ x 3

Order 4: $C_4$ x 6, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2048: $C_8\times C_{256}$

Order 1024: $C_8\times C_{128}$, $C_4\times C_{256}$

Order 512: $C_4\times C_{128}$ x 2, $C_2\times C_{256}$, $C_8\times C_{64}$

Order 256: $C_2\times C_{128}$ x 2, $C_4\times C_{64}$ x 2, $C_8\times C_{32}$, $C_{256}$

Order 128: $C_2\times C_{64}$ x 3, $C_4\times C_{32}$ x 2, $C_{128}$ x 2, $C_8\times C_{16}$

Order 64: $C_2\times C_{32}$ x 3, $C_{64}$ x 3, $C_4\times C_{16}$ x 2, $C_8^2$

Order 32: $C_{32}$ x 4, $C_2\times C_{16}$ x 3, $C_4\times C_8$ x 2

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

Order 8: $C_8$ x 4, $C_2\times C_4$ x 2

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2048: $C_8\times C_{256}$ ($C_1$)

Order 1024: $C_4\times C_{256}$ ($C_2$) x 2, $C_8\times C_{128}$ ($C_2$)

Order 512: $C_2\times C_{256}$ ($C_4$) x 4, $C_4\times C_{128}$ ($C_2^2$), $C_4\times C_{128}$ ($C_4$), $C_8\times C_{64}$ ($C_4$)

Order 256: $C_{256}$ ($C_8$) x 8, $C_2\times C_{128}$ ($C_2\times C_4$) x 2, $C_2\times C_{128}$ ($C_8$) x 2, $C_4\times C_{64}$ ($C_2\times C_4$), $C_4\times C_{64}$ ($C_8$), $C_8\times C_{32}$ ($C_8$)

Order 128: $C_{128}$ ($C_2\times C_8$) x 4, $C_{128}$ ($C_{16}$) x 4, $C_2\times C_{64}$ ($C_{16}$) x 2, $C_8\times C_{16}$ ($C_{16}$), $C_2\times C_{64}$ ($C_2\times C_8$), $C_2\times C_{64}$ ($C_4^2$), $C_4\times C_{32}$ ($C_2\times C_8$), $C_4\times C_{32}$ ($C_{16}$)

Order 64: $C_{64}$ ($C_{32}$) x 4, $C_2\times C_{32}$ ($C_{32}$) x 2, $C_{64}$ ($C_4\times C_8$) x 2, $C_{64}$ ($C_2\times C_{16}$) x 2, $C_2\times C_{32}$ ($C_4\times C_8$), $C_2\times C_{32}$ ($C_2\times C_{16}$), $C_4\times C_{16}$ ($C_2\times C_{16}$), $C_4\times C_{16}$ ($C_{32}$), $C_8^2$ ($C_{32}$)

Order 32: $C_{32}$ ($C_{64}$) x 4, $C_4\times C_8$ ($C_{64}$) x 2, $C_2\times C_{16}$ ($C_{64}$) x 2, $C_{32}$ ($C_2\times C_{32}$) x 2, $C_4\times C_8$ ($C_2\times C_{32}$), $C_2\times C_{16}$ ($C_2\times C_{32}$), $C_2\times C_{16}$ ($C_4\times C_{16}$), $C_{32}$ ($C_4\times C_{16}$), $C_{32}$ ($C_8^2$)

Order 16: $C_2\times C_8$ ($C_{128}$) x 4, $C_{16}$ ($C_{128}$) x 4, $C_{16}$ ($C_2\times C_{64}$) x 2, $C_2\times C_8$ ($C_2\times C_{64}$), $C_2\times C_8$ ($C_4\times C_{32}$), $C_4^2$ ($C_2\times C_{64}$), $C_{16}$ ($C_8\times C_{16}$), $C_{16}$ ($C_4\times C_{32}$)

Order 8: $C_8$ ($C_{256}$) x 8, $C_2\times C_4$ ($C_2\times C_{128}$) x 2, $C_8$ ($C_2\times C_{128}$) x 2, $C_2\times C_4$ ($C_4\times C_{64}$), $C_8$ ($C_4\times C_{64}$), $C_8$ ($C_8\times C_{32}$)

Order 4: $C_4$ ($C_2\times C_{256}$) x 4, $C_2^2$ ($C_4\times C_{128}$), $C_4$ ($C_4\times C_{128}$), $C_4$ ($C_8\times C_{64}$)

Order 2: $C_2$ ($C_4\times C_{256}$) x 2, $C_2$ ($C_8\times C_{128}$)

Order 1: $C_1$ ($C_8\times C_{256}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2048: $C_8\times C_{256}$ ($C_1$)

Order 1024: $C_8\times C_{128}$ ($C_2$), $C_4\times C_{256}$ ($C_2$)

Order 512: $C_2\times C_{256}$ ($C_4$), $C_4\times C_{128}$ ($C_2^2$), $C_4\times C_{128}$ ($C_4$), $C_8\times C_{64}$ ($C_4$)

Order 256: $C_2\times C_{128}$ ($C_2\times C_4$), $C_2\times C_{128}$ ($C_8$), $C_4\times C_{64}$ ($C_2\times C_4$), $C_4\times C_{64}$ ($C_8$), $C_8\times C_{32}$ ($C_8$), $C_{256}$ ($C_8$)

Order 128: $C_8\times C_{16}$ ($C_{16}$), $C_2\times C_{64}$ ($C_2\times C_8$), $C_2\times C_{64}$ ($C_4^2$), $C_2\times C_{64}$ ($C_{16}$), $C_4\times C_{32}$ ($C_2\times C_8$), $C_4\times C_{32}$ ($C_{16}$), $C_{128}$ ($C_2\times C_8$), $C_{128}$ ($C_{16}$)

Order 64: $C_2\times C_{32}$ ($C_4\times C_8$), $C_2\times C_{32}$ ($C_2\times C_{16}$), $C_2\times C_{32}$ ($C_{32}$), $C_4\times C_{16}$ ($C_2\times C_{16}$), $C_4\times C_{16}$ ($C_{32}$), $C_8^2$ ($C_{32}$), $C_{64}$ ($C_4\times C_8$), $C_{64}$ ($C_2\times C_{16}$), $C_{64}$ ($C_{32}$)

Order 32: $C_4\times C_8$ ($C_2\times C_{32}$), $C_4\times C_8$ ($C_{64}$), $C_2\times C_{16}$ ($C_2\times C_{32}$), $C_2\times C_{16}$ ($C_4\times C_{16}$), $C_2\times C_{16}$ ($C_{64}$), $C_{32}$ ($C_2\times C_{32}$), $C_{32}$ ($C_4\times C_{16}$), $C_{32}$ ($C_8^2$), $C_{32}$ ($C_{64}$)

Order 16: $C_2\times C_8$ ($C_2\times C_{64}$), $C_2\times C_8$ ($C_4\times C_{32}$), $C_2\times C_8$ ($C_{128}$), $C_4^2$ ($C_2\times C_{64}$), $C_{16}$ ($C_8\times C_{16}$), $C_{16}$ ($C_2\times C_{64}$), $C_{16}$ ($C_4\times C_{32}$), $C_{16}$ ($C_{128}$)

Order 8: $C_2\times C_4$ ($C_2\times C_{128}$), $C_2\times C_4$ ($C_4\times C_{64}$), $C_8$ ($C_2\times C_{128}$), $C_8$ ($C_4\times C_{64}$), $C_8$ ($C_8\times C_{32}$), $C_8$ ($C_{256}$)

Order 4: $C_2^2$ ($C_4\times C_{128}$), $C_4$ ($C_2\times C_{256}$), $C_4$ ($C_4\times C_{128}$), $C_4$ ($C_8\times C_{64}$)

Order 2: $C_2$ ($C_8\times C_{128}$), $C_2$ ($C_4\times C_{256}$)

Order 1: $C_1$ ($C_8\times C_{256}$)

Series

Derived series

$C_8\times C_{256}$

$\rhd$

$C_1$

Chief series

$C_8\times C_{256}$

$\rhd$

$C_4\times C_{256}$

$\rhd$

$C_4\times C_{128}$

$\rhd$

$C_2\times C_{128}$

$\rhd$

$C_2\times C_{64}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_8\times C_{256}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_8\times C_{256}$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

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

Character theory

Complex character table

The $2048 \times 2048$ character table is not available for this group.

Rational character table

The $62 \times 62$ rational character table is not available for this group.