C₂²×#2102 #6432444

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center	C₂²×C₄	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	#400
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2078 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

maximal subgroup of	C₂²×#27694, C₂²×#23834, C₂²×#29384
maximal normal subgroup of	C₂²×#27694, C₂²×#23834, C₂²×#29384

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339640);