C₅×#9991 #11932711

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#9991

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

698 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,10)(9,11)(12,13)(14,15,20,25)(16,26,30,37)(17,31,35,40)(18,24,19,23)(21,29,38,28)(22,33,41,34)(27,36,43,44)(32,45,42,39);(10,13)(14,16,20,30)(15,21,25,38)(17,32,35,42)(18,29,19,28)(22,39,41,45)(23,37,24,26)(27,34,43,33)(31,44,40,36);(6,8)(7,10)(9,12)(11,13)(14,17,20,35)(15,22,25,41)(16,27,30,43)(18,33,19,34)(21,36,38,44)(23,40,24,31)(26,39,37,45)(28,32,29,42)

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, 2979);