C₅×#7291 #11930011

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7291

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
socle	C₂²×C₅	#340
prime radical	C₂³×C₅	#106
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

192 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4,5)(6,7,9,12,10,13,17,20)(8,14,16,21,15,11,18,19)(23,27)(24,31)(25,32)(28,30)(29,36)(33,37)(34,35);(6,8)(7,11,13,14)(9,16)(10,15)(12,19,20,21)(17,18)(22,23,24,28,32,34,37,35,26,30,33,27,25,29,31,36)

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 279);