C₂²×#1621 #6348047

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

4958 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)(7,9)(8,11)(10,12)(14,17)(15,19)(18,20);(6,9)(7,12)(10,11)(15,19)(17,20);(13,14,16,18)(15,20,19,17)

maximal subgroup of	C₂²×#20138, C₂²×#20141, C₂²×#20272, …
maximal normal subgroup of	C₂²×#20138, C₂²×#20141, C₂²×#20272, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255243);