C₂×#15243 #608592

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰×C₄

Aliases

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

9078 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(8,10)(12,14)(17,21)(19,22);(4,6)(8,10)(11,12,13,14)(15,16,18,20)(17,19,21,22);(3,4,5,6)(7,8)(9,10)(11,12)(13,14);(15,17,18,21)(16,19,20,22)

maximal subgroup of	C₂×#1494235, C₂×#2921768, C₂×#649046, …
maximal normal subgroup of	C₂×#1494235, C₂×#2921768, C₂×#649046, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515788);