C₅×#9008 #11931728

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9008

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#120
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#386
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

398 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 118 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)(6,7,12,15,11,8,10,14)(9,13,19,21,18,16,17,20)(22,23)(24,33,36,44)(25,37)(26,31)(27,41)(28,32)(29,35,45,40)(30,38)(34,49,52,43)(39,46)(42,47)(48,50,53,51);(6,8,11,7)(9,16,18,13)(10,14,12,15)(17,20,19,21)(22,24,27,35,28,36,26,40)(23,29,41,33,32,45,31,44)(25,34,42,50,39,52,38,51)(30,43,37,48,47,49,46,53);(6,9,11,18)(7,13,8,16)(10,19,12,17)(14,21,15,20)(22,25,28,39)(23,30,32,47)(24,34,36,52)(26,38,27,42)(29,43,45,49)(31,46,41,37)(33,48,44,53)(35,50,40,51)

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, 1996);