C₅×#62849 #11985569

order	1280
nilpotency class	2
isoclinism	isoclinic class of #62842
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₅ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₅×#62849

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅
prime radical	C₂⁴×C₅	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

6578 conjugacy classes of subgroups; only the 6270 normal subgroups are shown.

normal subgroups only

This group has 6578 conjugacy classes of subgroups — too many to draw. Only its 6270 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 — 6 generators on 45 points:

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

Conjugacy classes & character table

410 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 55837);