C₂×#30447 #7500777

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4008 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);(3,4)(5,6)(11,12,15,17,16,14,13,18);(7,8)(9,10)(11,13)(12,14)(15,16);(4,6)(8,10)(11,13,16,15)(12,17,14,18);(4,6)(11,14,16,12)(13,18,15,17)

maximal subgroup of	C₂×#6742086, C₂×#6682668, C₂×#7504012, …
maximal normal subgroup of	C₂×#6742086, C₂×#6682668, C₂×#7504012, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7407973);