C₅×#2848 #10616443

order	640
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

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

1068 conjugacy classes of subgroups; only the 288 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 288 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) = 160

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18525);