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×#52516 |
| 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 | |
4984 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);(5,9)(8,10)(12,17)(13,15)(19,20)(21,26)(22,30)(23,28)(24,25)(27,31)(29,32)(33,34);(3,4)(5,8)(6,7)(9,10)(11,12,16,17)(13,14,15,18)(20,25)(21,28)(22,31)(32,33);(4,7)(8,10)(11,13,16,15)(12,18,17,14)(19,21,24,28)(20,26,25,23)(22,29,31,34)(27,33,30,32);(3,4)(5,8)(6,7)(9,10)(11,14,16,18)(12,13,17,15)(19,22)(20,27)(21,29)(23,32)(24,31)(25,30)(26,33)(28,34);(3,5)(4,8)(6,9)(7,10)(11,15,16,13)(12,18,17,14)(19,23,24,26)(20,28,25,21)(22,32,31,33)(27,34,30,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999546);