C₂×#10268 #232996

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂⁴	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1902 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 22 points:

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

maximal subgroup of	C₂×#249471, C₂×#249535, C₂×#244849, …
maximal normal subgroup of	C₂×#249471, C₂×#249535, C₂×#244849, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140192);