Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62885 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 34 points:
(1,2);(3,4)(5,9)(6,13)(7,8)(10,14)(11,15)(12,18)(16,17)(19,20,25,27)(21,31,32,26)(22,23,29,28)(24,34,33,30);(5,11)(9,15)(12,17)(16,18)(19,21)(20,26)(22,33)(23,34)(24,29)(25,32)(27,31)(28,30);(4,8)(9,15)(10,13)(16,18)(20,27)(21,32)(22,29)(30,34);(3,4)(5,9)(6,10)(7,8)(11,15)(12,16)(13,14)(17,18)(19,22,25,29)(20,28,27,23)(21,33,32,24)(26,30,31,34);(3,5,7,11)(4,9,8,15)(6,12,14,17)(10,16,13,18)(19,23,25,28)(20,29,27,22)(21,30,32,34)(24,26,33,31);(3,6)(4,10)(5,12)(7,14)(8,13)(9,16)(11,17)(15,18)(19,24,25,33)(20,30,27,34)(21,29,32,22)(23,26,28,31)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493415);