C₅×#2024 #10603532

order	640
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

290 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

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

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,3,4,5)(10,11,13,15)(12,16,17,14)(19,22)(20,24)(23,25);(6,7)(8,9)(10,12,13,17)(11,14,15,16);(7,9)(18,19,21,23)(20,25,24,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5614);