C₅×#19096 #11941816

order	1280
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#19096

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 37
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 37 points:

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

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, 12084);