C₅×#8509 #11931229

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8509

Subgroups

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

656 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 41
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,7)(8,9)(10,11,16,19,14,13,15,18)(12,17,23,25,21,20,22,24)(27,31,33,36)(32,40,41,39);(10,12,14,21)(11,17,13,20)(15,23,16,22)(18,25,19,24)(26,27)(28,32)(29,36)(30,33)(31,38)(34,39)(35,41)(37,40);(7,9)(10,13,14,11)(12,20,21,17)(15,18,16,19)(22,24,23,25)(26,28,30,35)(27,32,33,41)(29,37,38,34)(31,39,36,40)

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, 1497);