C₂×C₅×#393 #10615980

order	640
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁷

Aliases

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

1036 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 328 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2)(19,20,21,22,23);(8,10)(11,12,14,16)(13,17,18,15);(4,6)(8,10)(11,13,14,18)(12,15,16,17);(3,4)(5,6)(7,8)(9,10)(11,12)(13,17)(14,16)(15,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18062);