C₂²×#3246 #10575592

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
prime radical	C₂²×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

8090 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 20 points:

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

maximal subgroup of	C₂²×#61961, C₂²×#62096, C₂²×#60962
maximal normal subgroup of	C₂²×#61961, C₂²×#62096, C₂²×#60962

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10482788);