C₅×#12877 #11935597

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12877

Subgroups

	subgroup	G/H
center	C₄×C₅	#418
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1824
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

366 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7)(8,18,21,13)(9,23)(10,27)(11,33)(12,15)(14,25)(16,31)(17,30)(19,46,47,65)(20,50,28,60)(22,37,52,45)(24,55)(26,58)(29,43)(32,59)(34,42)(35,48,44,57)(36,53,41,62)(38,63)(39,56)(40,64)(49,67,54,69)(51,66,61,68);(6,8)(7,13)(9,19)(10,28)(11,22)(12,21)(14,35)(15,18)(16,41)(17,37)(20,32)(23,46)(24,47)(25,57)(26,49)(27,50)(29,61)(30,52)(31,62)(33,45)(34,54)(36,40)(38,66)(39,44)(42,69)(43,68)(48,56)(51,63)(53,64)(55,65)(58,67)(59,60);(6,9,11,34,12,24,30,26)(7,14,33,38,15,39,17,43)(8,19,22,54,21,47,52,49)(10,29,31,25,32,63,64,56)(13,35,45,66,18,44,37,68)(16,42,59,55,40,58,27,23)(20,51,53,48,28,61,62,57)(36,67,50,46,41,69,60,65)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5865);