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C₂²×#1282 #181067

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	#95
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7929, C₂²×#8965, C₂²×#8967
maximal normal subgroup of	C₂²×#7929, C₂²×#8965, C₂²×#8967
maximal quotient of	#72309243, #72309247

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88263);