C₅×#10509 #11933229

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10509

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 360 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) = 320

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

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3497);