C₂²×#1688 #6353612

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁴	#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2554 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);(6,9)(7,10)(11,12)(14,17)(15,19)(18,20);(5,6,10,12,8,11,7,9);(6,11)(9,12)(13,14,16,18)(15,20,19,17)

maximal subgroup of	C₂²×#20150, C₂²×#20272, C₂²×#19962
maximal normal subgroup of	C₂²×#20150, C₂²×#20272, C₂²×#19962

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260808);