C₂×C₅×#462 #10616628

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, 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

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

normal subgroups only

This group has 480 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) = 27
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2)(23,24,25,26,27);(4,6)(7,8,11,14)(9,15,17,22)(10,18,20,13)(12,19,21,16);(3,4,5,6);(7,9,11,17)(8,12,14,21)(10,19,20,16)(13,22,18,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18710);