C₂×#25123 #6438930

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

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

3790 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)(11,12,16,17,15,13,14,18);(4,6)(7,8)(9,10)(11,13,14,17,15,12,16,18);(8,10)(11,12,15,13)(14,17,16,18);(4,6)(11,14,15,16)(12,17,13,18)

maximal subgroup of	C₂×#6445075, C₂×#6575304, C₂×#6576754
maximal normal subgroup of	C₂×#6445075, C₂×#6575304, C₂×#6576754

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346126);