C₂²×#3384 #10576186

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

7766 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);(5,6,9,8)(7,11,12,10)(14,17)(15,19)(18,20);(5,7,9,12)(6,10,8,11)(14,18)(17,20);(13,14)(15,20)(16,18)(17,19);(5,8,9,6)(7,10,12,11)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10483382);