C₅×#13536 #11936256

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13536

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

318 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 320

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

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

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, 6524);