C₂×C₅×#450 #10616592

order	640
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

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

996 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 248 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) = 160

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18674);