C₅×#12502 #11935222

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12483
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12502

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

200 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 53
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7,10,13)(8,14,16,21)(9,18,15,12)(11,19,20,17)(22,23)(24,32)(25,28)(26,38)(27,41)(29,34)(30,40)(31,37)(33,48)(35,52)(36,49)(39,47)(42,46)(43,50)(44,51)(45,53);(6,8,15,19,10,16,9,17)(7,11,18,21,13,20,12,14)(23,28)(29,43)(30,44)(31,41)(32,48)(38,49)(45,47)(46,52);(8,16)(11,20)(14,21)(17,19)(22,24,25,33)(23,29,28,43)(26,39,36,53)(27,42,37,35)(30,46,44,52)(31,47,41,45)(32,49,48,38)(34,51,50,40)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5490);