Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29591 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,6,8)(5,9,10,7)(19,20,25,33,24,29,23,28)(21,26,30,34,22,32,31,27);(3,5,6,10)(4,7,8,9)(11,12,14,15)(13,16,17,18)(19,21)(20,26)(22,24)(23,31)(25,30)(27,28)(29,32)(33,34);(12,15)(16,18)(19,22,24,21)(20,27,29,34)(23,31,25,30)(26,33,32,28);(4,8)(7,9)(11,13,14,17)(12,16,15,18)(19,23,24,25)(20,28,29,33)(21,30,22,31)(26,34,32,27)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6369146);