C₂×C₅×#490 #10616728

order	640
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

488 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 200 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);(3,4)(5,8)(6,7)(9,10)(12,17)(13,18)(14,15);(4,7)(8,10)(11,12,14,17,16,18,15,13);(3,5,6,9)(4,8,7,10)(11,13,14,12,16,17,15,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18810);