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×#55568 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5038 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)(6,8)(7,10)(20,24)(21,27)(26,31)(28,33);(3,4,6,8)(5,7,10,9)(11,12,15,16)(13,14,17,18)(20,24)(25,30)(26,31)(32,34);(12,16)(13,17)(19,20,23,24)(21,25,27,30)(22,26,29,31)(28,32,33,34);(3,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,17)(16,18)(19,21,23,27)(20,25,24,30)(22,28,29,33)(26,32,31,34);(3,5,6,10)(4,7,8,9)(11,14)(12,13)(15,18)(16,17)(19,22,23,29)(20,26,24,31)(21,28,27,33)(25,32,30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006666);