C₅×#32801 #11955521

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₅×#32801

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

762 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 1280

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

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

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, 25789);