C₂×#128259 #72285352

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#128259

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂⁴	C₂×#97
prime radical	C₂⁵	#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2613 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 30 points:

(3,4)(5,6);(4,6)(7,8,10,12,11,13,14,9)(15,16,17,21,18,23,26,30,19,24,25,22,27,20,28,29);(1,2);(7,9,14,13,11,12,10,8)(16,20,24,23)(17,25)(21,30,22,29)(26,28)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72225051, #72286576, #72286765, …
maximal normal subgroup of	C₂×#72225051, #72286576, #72286765, …
maximal quotient of	C₂×#72217184, #72286765, C₂×#72217203, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4)(5,6), (4,6)(7,8,10,12,11,13,14,9)(15,16,17,21,18,23,26,30,19,24,25,22,27,20,28,29), (1,2), (7,9,14,13,11,12,10,8)(16,20,24,23)(17,25)(21,30,22,29)(26,28));

C2×#128259 #72285352