C₅×#13542 #11936262

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13542

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

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

normal subgroups only

This group has 252 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)(8,10)(9,11)(12,13)(14,15,17,22,18,23,30,20)(16,25,27,34,28,21,29,37)(19,24,31,39,32,40,45,35)(26,42,43,36,44,38,33,41);(6,8,9,12)(7,10,11,13)(15,20)(16,26,28,44)(17,30)(19,31)(21,36,25,41)(22,23)(27,33,29,43)(32,45)(34,38,37,42)(35,39);(7,11)(10,13)(14,16,17,27,18,28,30,29)(15,21,22,37,23,25,20,34)(19,33,31,26,32,43,45,44)(24,36,39,38,40,41,35,42)

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, 6530);