C₂×#10329 #234350

order	512
nilpotency class	3
isoclinism	isoclinic class of #10326
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2433	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3372 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

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

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

maximal subgroup of	C₂×#234400, #72231648
maximal normal subgroup of	C₂×#234400, #72231648

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141546);