C₅×#13689 #11936409

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13689

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₂×C₅×D₃₂	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

216 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 640

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

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

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6677);