Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32753 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12,14,17)(13,16,18,15)(19,20)(21,28)(22,31)(23,27)(24,26)(25,30)(29,34)(32,33);(3,5)(4,7)(6,9)(8,10)(11,13,14,18)(12,15,17,16)(21,29)(22,24)(25,33)(26,31)(28,34)(30,32);(4,8)(7,10)(11,13)(12,16)(14,18)(15,17)(19,21,23,30)(20,25,27,28)(22,29,24,32)(26,34,31,33);(5,9)(7,10)(13,18)(15,16)(19,22,23,24)(20,26,27,31)(21,29,30,32)(25,34,28,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538511);