C₅×#12885 #11935605

order	1280
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12885

Subgroups

	subgroup	G/H
center	C₅×C₈	#124
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1832
prime radical	C₂³×C₅	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ G

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7,11,17,12,18,35,50,13,19,36,51,10,16,32,47)(8,20,24,52,22,41,56,48,25,39,33,64,23,14,37,40)(9,26,30,59,29,46,62,49,31,44,34,65,28,15,42,45)(21,53,54,67,27,60,61,69,55,38,58,66,57,43,63,68);(6,8,10,23,13,25,12,22)(7,14,16,39,19,41,18,20)(9,27,29,55,31,57,28,21)(11,33,32,56,36,24,35,37)(15,43,26,53,46,60,44,38)(17,48,47,52,51,40,50,64)(30,63,62,54,34,61,42,58)(45,69,59,66,49,68,65,67);(6,9,10,28,13,31,12,29)(7,15,16,44,19,46,18,26)(8,21,22,27,25,55,23,57)(11,34,32,62,36,30,35,42)(14,38,20,43,41,53,39,60)(17,49,47,59,51,45,50,65)(24,58,56,63,33,54,37,61)(40,67,52,69,48,66,64,68)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5873);