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×#62953 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 34 points:
(1,2);(4,8)(9,15)(10,16)(17,18)(19,20,24,27)(21,22,29,25)(23,26,31,33)(28,30,34,32);(19,21,24,29)(20,25,27,22)(23,28,31,34)(26,32,33,30);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,21)(20,22)(23,28)(24,29)(25,27)(26,30)(31,34)(32,33);(3,5,7,12)(4,9,8,15)(6,13,14,11)(10,18,16,17)(19,22)(20,21)(23,30)(24,25)(26,28)(27,29)(31,32)(33,34);(3,6,7,14)(4,10,8,16)(5,11,12,13)(9,17,15,18);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,23,24,31)(20,26,27,33)(21,28,29,34)(22,30,25,32)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493493);