C₂²×#2547 #7628098

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

13690 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);(6,8)(14,16)(15,19);(9,10,11,12)(13,14,17,16)(15,18,19,20);(5,6,7,8)(9,10)(11,12)(13,15)(14,18)(16,20)(17,19);(13,16,17,14)(15,20,19,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535294);