C₅×#13366 #11936086

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13366

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

382 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 49
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5)(10,11,13,15)(12,16,17,14)(18,19,21,24)(20,28,29,46)(22,34,32,47)(23,37,33,27)(25,41,39,30)(26,43,40,36)(31,48,38,44)(35,45,49,42);(6,7)(8,9)(10,12,13,17)(11,14,15,16)(19,24)(25,39)(26,40)(27,37)(28,46)(34,47)(42,45)(44,48);(7,9)(18,20,21,29)(19,25,24,39)(22,35,32,49)(23,38,33,31)(26,44,40,48)(27,45,37,42)(28,47,46,34)(30,43,41,36)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6354);