C₂²×#3310 #10576252

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	C₂²×D₄²	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

9870 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10483448);