C₂²×#2065 #6432122

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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4682 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

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

maximal subgroup of	C₂²×#26834, C₂²×#23682, C₂²×#26490, …
maximal normal subgroup of	C₂²×#26834, C₂²×#23682, C₂²×#26490, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339318);