Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53001 |
| 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 | |
5326 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)(13,17)(16,18)(19,20,24,26)(21,27,29,22)(23,31,32,25)(28,30,34,33);(11,12)(13,16)(14,15)(17,18)(19,21,24,29)(20,22,26,27)(23,28,32,34)(25,30,31,33);(4,5)(8,9)(12,15)(16,18)(19,22,24,27)(20,21,26,29)(23,30,32,33)(25,28,31,34);(3,5,7,4)(6,9,10,8)(11,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,28)(24,29)(25,30)(26,27)(31,33)(32,34);(3,6,7,10)(4,8,5,9)(11,13,14,17)(12,16,15,18)(19,23,24,32)(20,25,26,31)(21,28,29,34)(22,30,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000044);