C₅×#9004 #11931724

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×#9004

Subgroups

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

458 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 41
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 41 points:

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

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, 1992);