C₅×#10084 #11932804

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

Aliases

direct product

C₅×#10084

Subgroups

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

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

normal subgroups only

This group has 484 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) = 1280

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

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

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, 3072);