Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C27 · Schur multiplier (H2) = C2 |
| direct product | C2×C4×C7⋊C27 |
| unambiguous semidirect product | (C2×C4×C7)⋊C27 |
| derived, Sylow | C7 ⋊ C2×C4×C27 |
| Hall | C2×C4×C7 ⋊ C27 |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 1512 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(8,9,10)(11,13,12)(14,15,18,16,19,24,21,26,31,17,20,25,22,27,32,29,34,37,23,28,33,30,35,38,36,39,40);(3,4,5,6)(7,8,9,11,10,12,13)360 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 19);