C₅×#2547 #10616142

order	640
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

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

604 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)(7,9)(15,17)(16,20);(10,11,12,13)(14,15,18,17)(16,19,20,21);(6,7,8,9)(10,11)(12,13)(14,16)(15,19)(17,21)(18,20);(14,17,18,15)(16,21,20,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18224);