Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11498 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4)(5,11,13,8)(6,9)(7,10)(12,17,18,15)(14,16)(19,20,24,30,26,29,39,47,25,31,38,48,23,40,22,28)(21,32,35,46,34,27,33,43,36,45,49,42,41,44,37,50);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,21,25,36)(20,27,31,44)(22,37,39,33)(23,41,26,34)(24,35,38,49)(28,46,47,42)(29,45,40,32)(30,43,48,50);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,22,23,38,25,39,26,24)(20,28,40,48,31,47,29,30)(21,33,41,35,36,37,34,49)(27,42,32,43,44,46,45,50)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387365);