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C₂²×#1592 #6344579

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

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

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#17963, C₂²×#18020, C₂²×#18022
maximal normal subgroup of	C₂²×#17963, C₂²×#18020, C₂²×#18022
maximal quotient of	C₂²×#9556

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251775);