C₅×#11491 #11934211

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11491

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₅×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

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

normal subgroups only

This group has 356 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) = 45
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 45 points:

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

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, 4479);