C₅×#12878 #11935598

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12878

Subgroups

	subgroup	G/H
center	C₄×C₅	#418
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1824
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

362 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 138 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)(6,7,11,18,13,16,12,35)(8,19,22,40,23,14,34,39)(9,24,27,56,25,57,36,46)(10,28,31,44,33,60,32,17)(15,30,41,58,43,64,42,26)(20,48,51,69,49,65,52,67)(21,53,55,45,29,61,63,38)(37,54,66,50,47,62,68,59);(6,8)(7,14)(9,20)(10,29)(11,34)(12,22)(13,23)(15,37)(16,19)(17,45)(18,40)(21,33)(24,48)(25,49)(26,59)(27,52)(28,53)(30,62)(31,55)(32,63)(35,39)(36,51)(38,44)(41,68)(42,66)(43,47)(46,69)(50,58)(54,64)(56,67)(57,65)(60,61);(6,9,12,36,13,25,11,27)(7,15,35,42,16,43,18,41)(8,20,22,51,23,49,34,52)(10,30,32,26,33,64,31,58)(14,37,39,66,19,47,40,68)(17,46,60,57,44,56,28,24)(21,54,55,50,29,62,63,59)(38,67,53,48,45,69,61,65)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5866);