C₅×#10517 #11933237

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10517

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

344 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 53
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7)(8,16)(9,14)(10,13)(11,15)(12,18)(17,21)(19,20)(22,23,27,31)(24,32,35,47)(25,36,39,29)(26,40,42,30)(28,41,44,34)(33,49,37,48)(38,46,53,51)(43,50,45,52);(6,8)(7,12)(9,18)(10,19)(11,17)(13,21)(14,16)(15,20)(22,24,27,35)(23,28,31,44)(25,37,39,33)(26,41,42,34)(29,45,36,43)(30,47,40,32)(38,50,53,52)(46,48,51,49);(7,13)(8,17)(10,11)(14,15)(18,19)(20,21)(22,25,26,38,27,39,42,53)(23,29,30,46,31,36,40,51)(24,33,34,50,35,37,41,52)(28,43,32,48,44,45,47,49)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3505);