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×#55645 |
| 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 | |
5374 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)(5,10)(6,9)(7,8)(12,15)(16,18)(19,20,24,27)(21,22,25,30)(23,26,32,34)(28,29,31,33);(3,5)(4,9)(6,8)(7,10)(13,17)(16,18)(19,21,24,25)(20,22,27,30)(23,31,32,28)(26,33,34,29);(5,6)(9,10)(11,12,14,15)(13,16,17,18)(19,21)(20,22)(23,28)(24,25)(26,29)(27,30)(31,32)(33,34);(3,6,8,5)(4,9,7,10)(11,12)(13,16)(14,15)(17,18)(19,22)(20,25)(21,27)(23,29)(24,30)(26,31)(28,34)(32,33);(3,7,8,4)(5,9,6,10)(11,13)(12,16)(14,17)(15,18)(19,23)(20,26)(21,28)(22,29)(24,32)(25,31)(27,34)(30,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008149);