Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #564⋊C5 |
| automorphism | Aut(G) = AΓL(1,𝔽16) (order 960) · Inn(G) = C24⋊C5 · Out(G) = Dic3 |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #564⋊C5 |
| Fitting, derived, Sylow | #564 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | C24⋊C5 |
| commutator, Fitting, cosocle, hypocenter | #564 | C5 |
| LCS | G ▷ #564 | |
| UCS | C1 ◁ C22 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(2,5,25,38,45)(3,10,35,27,20)(4,18,52,41,58)(8,31,39,63,37)(9,13,12,30,54)(11,29,59,19,34)(14,32,53,33,51)(15,48,17,42,43)(16,26,46,61,49)(22,23,44,47,55)(24,40,56,36,62)(28,57,60,50,64);(1,53,6,42)(2,29,11,5)(3,40,16,60)(4,36,23,64)(7,26,21,10)(8,15,13,51)(9,28,37,62)(12,39,31,30)(14,59,48,25)(17,18,32,44)(19,58,38,22)(20,54,49,63)(24,41,57,55)(27,45,61,34)(33,56,43,50)(35,47,46,52)Polycyclic presentation:
5,2^6|b^2=g,c^2=f,d^2=fg,e^2=g,b^a=e,c^a=be,d^a=bce,e^a=bcde,c^b=cf,d^b=dfg,e^b=ef,d^c=dfg,e^c=eg,e^d=eg23 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1012);