C₂²×#2217 #7622958

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂²×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2258 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2);(3,4);(5,6)(7,8)(10,14)(11,18)(12,19)(15,20)(16,24)(17,23)(21,22);(9,10,15,24,12,16,11,14,13,17,18,22,19,21,20,23);(6,8)

maximal subgroup of	C₂²×#33746, C₂²×#33873, C₂²×#33830, …
maximal normal subgroup of	C₂²×#33746, C₂²×#33873, C₂²×#33830, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530154);