C₅×#13534 #11936254

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13534

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

330 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(10,11,18,22,25,40,14,21,17,24,15,31,16,23,30,34)(12,20,27,37,33,35,26,36,29,39,13,19,28,38,32,41);(6,7)(8,9)(10,12,25,28,17,29,16,33)(11,19,23,41,24,37,40,36)(13,18,26,14,27,15,32,30)(20,34,35,31,39,21,38,22);(7,9)(10,13)(11,20)(12,18)(14,28)(15,29)(16,32)(17,27)(19,34)(21,37)(22,36)(23,35)(24,39)(25,26)(30,33)(31,41)(38,40)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6522);