C₂×#24269 #6442746

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#2408	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5500 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)(12,16)(13,18)(14,17);(7,8)(9,10)(11,12,14,16,15,13,17,18);(4,6)(8,10)(12,13)(16,18);(11,13,14,18,15,12,17,16)

maximal subgroup of	C₂×#6573480, C₂×#6442768, C₂×#6573500, …
maximal normal subgroup of	C₂×#6573480, C₂×#6442768, C₂×#6573500, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349942);