C₅×#12351 #11935071

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12351

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂×C₅×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#436 ◁ G

286 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 640

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

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5339);