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×#62879 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31482 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)(19,20,23,26)(21,24,29,27)(22,30,31,25)(28,33,34,32);(5,11)(9,15)(12,17)(16,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,31)(32,34);(4,8)(9,15)(10,13)(16,18)(19,21)(20,24)(22,28)(23,29)(25,32)(26,27)(30,33)(31,34);(3,4)(5,9)(6,10)(7,8)(11,15)(12,16)(13,14)(17,18)(19,22,23,31)(20,25,26,30)(21,28,29,34)(24,32,27,33);(3,5,7,11)(4,9,8,15)(6,12,14,17)(10,16,13,18);(3,6)(4,10)(5,12)(7,14)(8,13)(9,16)(11,17)(15,18)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493409);