Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29489 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2144 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)(5,9)(6,7)(8,10)(11,12)(13,17)(14,16)(15,18)(19,20,24,26,22,27,23,28)(21,29,32,34,30,25,31,33);(4,7)(8,9)(11,13)(12,15)(14,18)(16,17)(19,21)(20,25)(22,30)(23,31)(24,32)(26,33)(27,29)(28,34);(13,18)(15,17)(20,26)(23,24)(25,33)(27,28)(29,34)(31,32);(3,5,6,10)(4,8,7,9)(12,16)(15,17)(21,30)(25,29)(31,32)(33,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371436);