C₂²×#1626 #6348033

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3366 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);(6,10)(7,8)(11,12)(14,18)(15,19)(16,20);(5,6)(7,10)(8,12)(9,11)(13,14,17,15)(16,19,20,18);(5,7,9,8)(6,10,11,12)(13,15,17,14)(16,18,20,19)

maximal subgroup of	C₂²×#20272
maximal normal subgroup of	C₂²×#20272

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255229);