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×#55870 |
| 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 | |
5310 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)(11,12)(13,15)(14,17)(16,18)(19,20,24,27)(21,22,29,25)(23,26,31,33)(28,30,34,32);(3,4,6,8)(5,7,10,9)(12,13)(16,17)(23,31)(26,33)(28,34)(30,32);(11,12,15,13)(14,16,18,17)(19,21,24,29)(20,25,27,22)(23,28,31,34)(26,32,33,30);(3,4)(5,7)(6,8)(9,10)(11,13,15,12)(14,17,18,16)(19,22,24,25)(20,21,27,29)(23,30,31,32)(26,28,33,34);(3,5,6,10)(4,7,8,9)(11,14,15,18)(12,16,13,17)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008126);