Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | Q8×#562 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#562 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8);(10,13)(11,15)(17,18)(19,23)(20,21)(22,24);(10,13)(14,16)(18,21)(22,23);(9,10)(11,14)(12,13)(15,16)(17,19,20,24)(18,22,21,23);(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999015);