C₂×#28458 #6465607

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4598 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 18 points:

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

maximal subgroup of	C₂×#6585687, C₂×#6465861, C₂×#7521471, …
maximal normal subgroup of	C₂×#6585687, C₂×#6465861, C₂×#7521471, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372803);