Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | Aut(G) = C2×#66297 (order 640) · Inn(G) = C24⋊C5 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #585⋊C5 |
| Fitting, Sylow | #585 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C24⋊C5 |
| commutator, cosocle, hypocenter | 2−1+4 | C2×C5 |
| Frattini, socle | C2 | C2×C24⋊C5 |
| Fitting | #585 | C5 |
| LCS | G ▷ 2−1+4 | |
| UCS | C1 ◁ C4 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,7,12)(3,11,54,38,17,8,28,64,14,33,16,44,26,30,13,32,6,42,22,29)(4,31,45,60,23,9,27,55,19,39,20,43,46,35,18,36,15,56,50,34)(5,37,49,62,53,10,48,59,47,63,25,58,51,57,24,41,21,61,52,40);(1,27,7,15)(2,43,12,31)(3,28,16,6)(4,21,20,48)(5,47,25,52)(8,44,32,11)(9,37,36,58)(10,57,41,62)(13,19,17,50)(14,24,22,53)(18,54,23,26)(29,35,33,60)(30,40,38,63)(34,64,39,42)(45,51,46,49)(55,61,56,59)Polycyclic presentation:
2,5,2^5|a^2=g,d^2=g,e^2=g,c^b=f,d^b=cf,e^b=cdf,f^b=cdef,d^c=dg,e^c=eg,f^c=fg,e^d=eg26 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1586);