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C₂×#12605 #492949

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂×#97
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(3,4)(5,6)(8,13,14,16)(10,18)(15,22,21,20)(17,19);(4,6)(7,8,10,15,11,16,19,22,12,14,18,21,9,13,17,20);(8,13,14,16)(10,18)(15,22,21,20)(17,19)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#493177, #72216446, #72216453, …
maximal normal subgroup of	C₂×#493177, #72216446, #72216453, …
maximal quotient of	#72217246, D₃×#12605, #72231333, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400145);