C₅×#12423 #11935143

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12423

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

218 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 1280

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

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

Conjugacy classes & character table

215 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5411);