Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#47264 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,14,16)(13,15,17,18)(19,20)(21,22)(23,31)(24,25)(26,29)(27,32)(28,33)(30,34);(3,5,7,10)(4,8,6,9)(20,25)(22,26)(27,31)(30,33);(5,10)(8,9)(11,12)(13,15)(14,16)(17,18)(19,21)(20,26)(22,25)(23,28)(24,29)(27,33)(30,31)(32,34);(3,5)(4,9)(6,8)(7,10)(11,13,14,17)(12,15,16,18)(19,22,24,26)(20,21,25,29)(23,30,32,33)(27,28,31,34);(3,6,7,4)(5,8,10,9)(19,23,24,32)(20,27,25,31)(21,28,29,34)(22,30,26,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608570);