Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
| direct product | C8×#563 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C24×C4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,3,4,5,6,7,8);(10,13)(11,15)(17,18,22,20)(19,25,27,23)(21,28,30,24)(26,31,29,32);(10,13)(14,16)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31);(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,23,27,25)(21,28,30,24)(26,31,29,32);(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315185);