C₅×#11468 #11934188

order	1280
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11468

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×#525	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂³×C₅×C₈ ◁ G

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

normal subgroups only

This group has 398 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) = 320

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

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

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, 4456);