C₅×#8533 #11931253

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8533

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

656 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 320

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

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1521);