C₅×#12513 #11935233

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₅×#12513

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

290 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 640

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

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

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, 5501);