C₂²×#2066 #6432124

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3842 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

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

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

maximal subgroup of	C₂²×#24738, C₂²×#26842, C₂²×#26340, …
maximal normal subgroup of	C₂²×#24738, C₂²×#26842, C₂²×#26340, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339320);