Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#58809 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5292 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)(5,7)(6,9)(8,10)(11,12)(13,14)(15,16)(17,18)(19,20,25,22)(21,26,23,27)(24,28,32,30)(29,33,31,34);(4,5)(6,10)(12,16)(13,17)(19,21,25,23)(20,26,22,27)(24,31,32,29)(28,34,30,33);(4,5)(8,9)(19,22,25,20)(21,26,23,27)(24,30,32,28)(29,33,31,34);(3,5,7,4)(6,9,10,8)(11,13,15,17)(12,14,16,18)(19,23,25,21)(20,27,22,26)(24,31,32,29)(28,34,30,33);(3,6,7,10)(4,8,5,9)(11,14)(12,17)(13,16)(15,18)(19,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,33)(27,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012299);