C₂×#25791 #6443220

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
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×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6116 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

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

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

maximal subgroup of	C₂×#6443244, C₂×#6573500, C₂×#6573487
maximal normal subgroup of	C₂×#6443244, C₂×#6573500, C₂×#6573487

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350416);