C₂×C₅×#475 #10616668

order	640
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

416 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 640

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18750);