Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#24988 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9)(5,13)(6,18)(7,19)(11,26)(12,23)(16,17)(20,31)(21,25)(24,32)(27,29)(28,34);(3,4,7,9,8,12,19,23)(5,14,13,21,17,10,16,25)(6,11,18,22,20,26,31,33)(15,28,27,32,30,24,29,34);(3,5)(4,10)(6,15)(7,16)(8,17)(9,21)(11,24)(12,14)(13,19)(18,29)(20,30)(22,32)(23,25)(26,28)(27,31)(33,34);(3,6,19,31,8,20,7,18)(4,11,23,33,12,26,9,22)(5,15,13,27,17,30,16,29)(10,24,25,34,14,28,21,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354245);