C₅×#2031 #10603539

order	640
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center	C₄×C₅	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅
socle	C₂×C₅	#393
prime radical	C₂×C₅×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ G

244 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5621);