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×#56686 |
| 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 | |
5000 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)(6,8)(9,10)(11,12,16,14)(13,17,15,18)(19,20)(21,27)(22,30)(23,24)(25,29)(26,31)(28,33)(32,34);(6,10)(8,9)(12,14)(13,15)(20,23)(22,31)(25,27)(28,32);(4,5)(8,9)(11,13,16,15)(12,17,14,18)(19,21)(20,25)(22,32)(23,27)(24,29)(26,34)(28,31)(30,33);(3,5,7,4)(6,9,10,8)(11,14,16,12)(13,17,15,18)(19,22)(20,26)(21,28)(23,30)(24,31)(25,33)(27,34)(29,32);(3,6)(4,8)(5,9)(7,10)(11,15,16,13)(12,17,14,18)(19,23,24,20)(21,27,29,25)(22,30,31,26)(28,34,32,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006623);