Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35511 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5160 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)(11,12,16,15)(13,18,14,17)(19,20)(21,25)(22,26)(23,24)(27,31)(28,33)(29,34)(30,32);(3,4,7,6)(5,8,9,10)(11,13,16,14)(12,17,15,18)(20,24)(21,27)(26,32)(28,29);(3,5)(4,8)(6,10)(7,9)(11,14,16,13)(12,18,15,17)(19,21,23,27)(20,25,24,31)(22,29,30,28)(26,34,32,33);(11,12,16,15)(13,17,14,18)(19,22,23,30)(20,26,24,32)(21,28,27,29)(25,33,31,34);(3,6,7,4)(5,10,9,8)(11,15,16,12)(13,18,14,17)(19,20)(21,25)(22,26)(23,24)(27,31)(28,33)(29,34)(30,32)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536083);