C₅×#32786 #11955506

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#32786

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 640

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

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 25774);