C₅×#12500 #11935220

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

Aliases

direct product

C₅×#12500

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

212 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 1280

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 5488);