C₂×C₅×#392 #10615979

order	640
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁸

Aliases

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

1812 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.

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

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18061);