C₅×#12066 #11934786

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12066

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#442
commutator	C₈	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#322 ◁ G

214 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 45
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,7,9,12,10,8,11,13)(14,15,18,24,20,25,19,22)(16,26,28,43,30,44,29,40)(17,31,34,41,33,39,35,23)(21,36,37,27,38,32,42,45);(14,16,20,30)(15,21,25,38)(17,32,33,36)(18,29,19,28)(22,37,24,42)(23,40,41,43)(26,31,44,39)(27,35,45,34);(6,8,9,13,10,7,11,12)(15,22,25,24)(17,33)(21,37,38,42)(23,31,41,39)(26,40,44,43)(27,45)(32,36)(34,35)

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, 5054);