C₅×#13485 #11936205

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13485

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

314 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,11)(9,10)(12,13)(14,15,20,22,19,25,17,23)(16,21,29,37,30,38,27,35)(18,24,33,39,32,42,31,40)(26,41,43,34,44,45,28,36);(7,10)(11,13)(14,16)(15,21)(17,27)(18,26)(19,30)(20,29)(22,37)(23,35)(24,41)(25,38)(28,31)(32,44)(33,43)(34,39)(36,40)(42,45);(6,8,9,12)(7,11,10,13)(15,22)(16,26)(17,20)(18,31)(21,34)(23,25)(27,43)(28,29)(30,44)(32,33)(35,45)(36,38)(37,41)(39,40)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6473);