Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#23847 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #401 |
| commutator, Frattini, cosocle | C24 | C25 |
| socle | C22 | C2×#401 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1694 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)(5,13,16,9)(6,10,19,25)(7,11)(8,12)(14,23)(15,27,29,22)(17,21)(18,24,31,34)(20,26)(28,33)(30,32);(3,5)(4,9)(6,17,19,14)(7,15)(8,16)(10,23,25,21)(11,22)(12,13)(18,30,31,28)(20,29)(24,33,34,32)(26,27);(3,6)(4,10,12,25)(5,14)(7,18)(8,19)(9,21,13,23)(11,24,26,34)(15,28)(16,17)(20,31)(22,32,27,33)(29,30);(3,7,8,20)(4,11,12,26)(5,15,16,29)(6,18,19,31)(9,22,13,27)(10,24,25,34)(14,28,17,30)(21,32,23,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340006);