C₅×#11490 #11934210

order	1280
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11490

Subgroups

	subgroup	G/H
center	C₂×C₅×C₈	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×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

364 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 138 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)(14,15,19,25,21,24,33,41,20,26,32,42,18,34,17,23)(16,27,30,37,29,22,36,38,31,40,44,45,35,39,28,43);(7,10)(8,12)(14,16)(15,22)(17,28)(18,35)(19,30)(20,31)(21,29)(23,37)(24,40)(25,38)(26,39)(27,34)(32,44)(33,36)(41,45)(42,43);(14,17,18,32,20,33,21,19)(15,23,34,42,26,41,24,25)(16,28,35,44,31,36,29,30)(22,37,27,43,39,45,40,38)

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4478);