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C₂²×#1279 #180957

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center	C₂⁴×C₄	D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#108795, C₂×#110284, C₂²×#8170, …
maximal normal subgroup of	C₂×#108795, C₂×#110284, C₂²×#8170, …
maximal quotient of	C₂²×#7178

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88153);