C₅×#1721 #10603229

order	640
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₈

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

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

normal subgroups only

This group has 242 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) = 320

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5311);