C₂²×#2077 #6432173

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂⁵	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁶	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4162 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);(9,10)(11,14)(12,13)(15,16)(17,18,21,22,20,24,19,23);(5,6)(7,8)(10,13)(14,16);(6,8)(9,11,12,15)(10,14,13,16)(18,22)(19,21)(23,24)

maximal subgroup of	C₂²×#26130, C₂²×#24911, C₂²×#26128, …
maximal normal subgroup of	C₂²×#26130, C₂²×#24911, C₂²×#26128, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339369);