Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#11643 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
734 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)(5,12)(6,10)(7,16)(8,11)(9,14)(13,17)(15,18)(19,20,23,29,24,30,38,47,25,27,39,43,22,28,36,45)(21,31,33,48,32,41,49,46,34,44,37,50,35,26,40,42);(3,5,8,15)(4,9,11,17)(6,14,7,13)(10,18,16,12)(19,21)(20,26)(22,35)(23,37)(24,32)(25,34)(27,41)(28,44)(29,46)(30,31)(33,39)(36,49)(38,40)(42,43)(45,48)(47,50);(3,6,8,7)(4,10,11,16)(5,13,15,14)(9,12,17,18)(20,27)(26,41)(28,30)(29,43)(31,44)(42,46)(45,47)(48,50)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388343);