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C₂×#7025 #96226

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

Subgroups

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

1786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213034, C₂×#102905, #72214372, …
maximal normal subgroup of	#72213034, C₂×#102905, #72214372, …
maximal quotient of	C₂×#93386, C₂×#93376, #72211371, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3422);