C₅×#2044 #10603552

order	640
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	#120
commutator, Frattini, cosocle	C₂⁴	C₂³×C₅
prime radical	C₅×#534	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁴×C₅ ◁ G

382 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

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

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5634);