C₅×#32874 #11955594

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32873
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₅×#32874

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2840
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂⁴×C₅ ◁ G

688 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 160

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 25862);