C₅×#12496 #11935216

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12483
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#12496

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

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

normal subgroups only

This group has 234 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)(7,10)(8,11)(12,13)(14,15)(16,24)(17,20)(18,30)(19,33)(21,26)(22,32)(23,29)(25,40)(27,44)(28,41)(31,39)(34,38)(35,42)(36,43)(37,45);(6,7,11,13,9,12,8,10)(15,20)(21,35)(22,36)(23,33)(24,40)(30,41)(37,39)(38,44);(7,12)(10,13)(14,16,17,25)(15,21,20,35)(18,31,28,45)(19,34,29,27)(22,38,36,44)(23,39,33,37)(24,41,40,30)(26,43,42,32)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5484);