C₅×#12851 #11935571

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12851

Subgroups

	subgroup	G/H
center	C₄×C₅	#381
commutator	C₂³	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1817
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₅×C₈ ◁ G

416 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 3 generators on 37 points:

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

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5839);