C₅×#2452 #10616047

order	640
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

620 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18129);