Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52574 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4924 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);(5,9)(8,10)(11,12,15,17)(13,18,16,14)(19,20)(21,22)(23,27)(24,25)(26,29)(28,30)(31,32)(33,34);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(20,25)(22,26)(27,32)(30,33);(4,7)(8,10)(11,12,15,17)(13,14,16,18)(19,21)(20,26)(22,25)(23,28)(24,29)(27,33)(30,32)(31,34);(3,4)(5,8)(6,7)(9,10)(11,12)(13,18)(14,16)(15,17)(19,22,24,26)(20,21,25,29)(23,30,31,33)(27,28,32,34);(3,5)(4,8)(6,9)(7,10)(11,14,15,18)(12,16,17,13)(19,23,24,31)(20,27,25,32)(21,28,29,34)(22,30,26,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999528);