C₅×#2065 #10603573

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

284 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 320

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5655);