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×#57543 |
| 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 | |
5054 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);(4,7)(5,9)(20,24)(21,27)(26,31)(29,33);(3,4,6,7)(5,8,9,10)(11,12,15,17)(13,14,18,16)(22,30)(26,31)(29,33)(32,34);(3,4)(5,8)(6,7)(9,10)(11,13,15,18)(12,16,17,14)(19,20,23,24)(21,28,27,25)(22,26,30,31)(29,34,33,32);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,18)(16,17)(19,21,23,27)(20,25,24,28)(22,29,30,33)(26,32,31,34);(3,5)(4,8)(6,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,22,23,30)(20,26,24,31)(21,29,27,33)(25,32,28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007561);