C₂×C₅×#480 #10616680

order	640
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

512 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 216 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)(11,12)(13,15)(14,17)(16,18);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(12,15)(13,18)(16,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18762);