C₂×#24370 #6437657

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

5802 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);(3,4)(5,6)(7,8,9,10);(4,6)(11,12)(13,17)(14,16)(15,18);(8,10)(12,16)(13,14)(17,18);(4,6)(11,13,15,14)(12,17,18,16)

maximal subgroup of	C₂×#6437684, C₂×#6573502, C₂×#6573487
maximal normal subgroup of	C₂×#6437684, C₂×#6573502, C₂×#6573487

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344853);