C₅×#1651 #10603159

order	640
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₅
prime radical	C₂×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

234 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 76 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)(7,10)(8,12)(11,13)(14,15,18,21)(16,19,17,20);(6,7,9,11)(8,13,12,10)(15,19)(16,17)(20,21);(15,20)(16,17)(19,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5241);