C₂×#10564 #239351

order	512
nilpotency class	3
isoclinism	isoclinic class of #10564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3186	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1560 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 18 points:

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

maximal subgroup of	C₂×#267371, #72313406, #72313431, …
maximal normal subgroup of	C₂×#267371, #72313406, #72313431, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146547);