Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18422 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3188 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3104 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)(5,15)(6,9)(7,13)(8,22)(10,14)(11,17)(12,24)(16,25)(18,28)(19,31)(20,33)(21,26)(23,27)(29,34)(30,32);(3,5)(4,11)(6,16)(7,17)(8,25)(9,19)(10,20)(12,28)(13,31)(14,30)(15,22)(18,27)(21,33)(23,29)(24,32)(26,34);(3,6,9,4)(5,16,19,11)(7,12,8,21)(10,23,27,14)(13,26,22,24)(15,32,31,34)(17,28,25,33)(18,30,20,29);(3,7,9,8)(4,12,6,21)(5,17,19,25)(10,24,27,26)(11,28,16,33)(13,23,22,14)(15,30,31,29)(18,34,20,32)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254209);