C₂²×C₅×#124 #10619221

order	640
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

C₂²×C₅×#124

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅
prime radical	C₂³×C₅×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

1880 conjugacy classes of subgroups; only the 952 normal subgroups are shown.

normal subgroups only

This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 952 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 — 5 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 21303);