C₂×C₅×#533 #10619228

order	640
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center	C₂²×C₄×C₅	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅
socle	C₂³×C₅	C₂⁴
prime radical	C₂⁴×C₅	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

1168 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 5 generators on 23 points:

(1,2)(19,20,21,22,23);(3,4)(5,8)(6,7)(9,10)(11,12,14,16)(13,17,18,15);(4,7)(8,10);(11,13,14,18)(12,15,16,17);(3,5,6,9)(4,8,7,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 21310);