C₂×#10290 #234113

order	512
nilpotency class	3
isoclinism	isoclinic class of #10280
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#2867	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2758 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)(14,17)(16,18);(4,7)(5,10)(8,9)(13,18)(15,16);(5,10)(7,8)(12,15)(13,18)(16,17)

maximal subgroup of	C₂×#247880, C₂×#234216, C₂×#248071, …
maximal normal subgroup of	C₂×#247880, C₂×#234216, C₂×#248071, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141309);