C₅×#10083 #11932803

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10083

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

526 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 640

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

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

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, 3071);