C₅×#10529 #11933249

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10529

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 640

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3517);