Abstract group 1024.dkh: $C_2^2.C_2^8$

• Label: 1024.dkh
• Order: \( 2^{10} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{8} \)
• $\card{Z(G)}$: 64
• $\card{\Aut(G)}$: \( 2^{41} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{37} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $8$

Group information

Description:	$C_2^2.C_2^8$
Order:	\(1024\)\(\medspace = 2^{10} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	Group of order \(692692325498880\)\(\medspace = 2^{41} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 10
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	4
Elements	1	575	448	1024
Conjugacy classes	1	255	144	400
Divisions	1	255	144	400
Autjugacy classes	1	5	3	9

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$8$
Inequivalent generating 8-tuples:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{4}=c^{2}=d^{4}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $32$ $\langle(5,9)(12,15)(19,21)(27,30), (1,3,11,25)(2,4,13,26)(5,19)(6,20,16,14)(7,22,28,31) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^2$ . $C_2^8$	$(D_4\times C_2^6)$ . $C_2$ (4)	$(C_2^6:D_4)$ . $C_2$ (4)	$C_4$ . $(D_4\times C_2^5)$ (64)	all 14

Elements of the group are displayed as permutations of degree 32.

Homology

Abelianization:	$C_{2}^{8} $
Schur multiplier:	$C_{2}^{28}$
Commutator length:	$1$

Subgroups

There are 472117 normal subgroups (5 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2^6$
Commutator:	a subgroup isomorphic to $C_2^2$
Frattini:	a subgroup isomorphic to $C_2^2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2.C_2^8$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 2 (order at least 512) are shown, as well as normal subgroups of index up to 2 or of order up to 4.

Order 1024: $C_2^2.C_2^8$

Order 512: $C_2^3\times D_4^2$ x 240, $D_4\times C_2^6$ x 4, $C_2^4\times C_2^2:D_4$ x 4, $C_2^6:D_4$ x 4, $C_4^2.C_2^5$ x 2, $C_4^2.C_2^5$

Order 4: $C_2^2$ x 779, $C_4$ x 64

Order 2: $C_2$ x 63

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

All normal subgroups of index up to 2 or order up to 4 are shown.

Order 1024: $C_2^2.C_2^8$ ($C_1$)

Order 512: $C_2^3\times D_4^2$ ($C_2$) x 240, $D_4\times C_2^6$ ($C_2$) x 4, $C_2^4\times C_2^2:D_4$ ($C_2$) x 4, $C_2^6:D_4$ ($C_2$) x 4, $C_4^2.C_2^5$ ($C_2$) x 2, $C_4^2.C_2^5$ ($C_2$)

Order 256: ? (unidentified group of order 4) x 10795

Order 128: ? (unidentified group of order 8) x 97283

Order 64: ? (unidentified group of order 16) x 204819

Order 32: ? (unidentified group of order 32) x 118003

Order 16: ? (unidentified group of order 64) x 33491

Order 8: ? (unidentified group of order 128) x 6563

Order 4: $C_2^2$ ($C_2^2\times D_4^2$) x 560, $C_2^2$ ($D_4\times C_2^5$) x 188, $C_4$ ($D_4\times C_2^5$) x 64, $C_2^2$ ($D_4:C_2^5$) x 30, $C_2^2$ ($C_2^8$)

Order 2: $C_2$ ($C_2^3\times D_4^2$) x 60, $C_2$ ($D_4\times C_2^6$) x 2, $C_2$ ($D_4.C_2^6$)

Order 1: $C_1$ ($C_2^2.C_2^8$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $400 \times 400$ rational character table is not available for this group.