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C₂×#12544 #492836

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

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

1994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#147283, C₂×#147446, C₂×#146306, …
maximal normal subgroup of	C₂×#147283, C₂×#147446, C₂×#146306, …
maximal quotient of	#72322932

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400032);