Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C2×#20612 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×#379 |
| prime radical | C22×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17142 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);(3,4,7,12,8,11,20,25)(5,13,16,26,15,9,19,22)(6,10,18,24,17,23,30,33)(14,21,29,32,28,27,31,34);(3,5)(4,9)(6,14)(7,19)(8,15)(10,21)(11,13)(12,26)(16,20)(17,28)(18,31)(22,25)(23,27)(24,34)(29,30)(32,33);(3,6)(4,10)(5,14)(7,18)(8,17)(9,21)(11,23)(12,24)(13,27)(15,28)(16,29)(19,31)(20,30)(22,32)(25,33)(26,34);(4,11)(9,13)(10,23)(12,25)(21,27)(22,26)(24,33)(32,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277934);