C₅×#10667 #11933387

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10667

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#399
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 354 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) = 45
transitive degree	μ_t(G) = 320

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

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

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, 3655);