Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#16764 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4494 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);(4,7)(5,9)(11,12,14,16)(13,17,18,15)(19,20,23,24)(21,25,26,22)(28,31)(29,33);(4,7)(8,10)(20,24)(21,26)(27,28,30,31)(29,32,33,34);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,25)(22,24)(23,26)(27,28)(29,34)(30,31)(32,33);(3,5,6,9)(4,8,7,10)(11,13,14,18)(12,15,16,17)(19,22,23,25)(20,26,24,21)(27,29)(28,32)(30,33)(31,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518696);