C₂²×#1639 #6348024

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
prime radical	C₂²×D₄²	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

6242 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

maximal subgroup of	C₂²×#18529, C₂²×#18754, C₂²×#18755, …
maximal normal subgroup of	C₂²×#18529, C₂²×#18754, C₂²×#18755, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255220);