Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62910 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32484 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)(5,12)(6,9)(7,10)(8,14)(11,16)(13,17)(15,18)(21,28)(25,32)(29,33)(30,34);(3,5,9,7)(4,10,6,12)(8,13,16,15)(11,17,14,18)(19,20,24,27)(21,25,28,32)(22,23,26,31)(29,30,33,34);(3,4,9,6)(5,12,7,10)(8,11,16,14)(13,18,15,17)(19,21)(20,25)(22,29)(23,30)(24,28)(26,33)(27,32)(31,34);(3,6,9,4)(5,12,7,10)(8,14,16,11)(13,18,15,17)(19,22)(20,23)(21,29)(24,26)(25,30)(27,31)(28,33)(32,34);(3,7,9,5)(4,10,6,12)(8,15,16,13)(11,17,14,18)(19,23)(20,26)(21,30)(22,27)(24,31)(25,33)(28,34)(29,32);(3,8)(4,11)(5,13)(6,14)(7,15)(9,16)(10,17)(12,18)(19,21)(20,25)(22,29)(23,30)(24,28)(26,33)(27,32)(31,34)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493447);