C₂²×#3503 #10585974

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁷
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

36298 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 7 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493170);