C₂×#10230 #233929

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂⁴	#120
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

1926 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);(8,11)(9,13)(12,14)(16,19)(17,21)(20,22);(3,4)(5,6)(7,8,10,12)(9,14,13,11);(4,6)(15,16,18,20)(17,22,21,19)

maximal subgroup of	C₂×#249471, C₂×#244849, C₂×#233944, …
maximal normal subgroup of	C₂×#249471, C₂×#244849, C₂×#233944, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141125);