C₅×#13515 #11936235

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13515

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

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

normal subgroups only

This group has 298 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) = 33
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 33 points:

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

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, 6503);