C₅×#11469 #11934189

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11469

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

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

normal subgroups only

This group has 386 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) = 320

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

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

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, 4457);