Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54122 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(12,16)(14,18)(19,20,24,28)(21,25,29,34)(22,26,32,31)(23,27,33,30);(6,10)(8,9)(11,12,15,16)(13,18,17,14)(21,29)(23,33)(25,34)(27,30);(4,5)(8,9)(12,16)(13,17)(19,21)(20,25)(22,27)(23,31)(24,29)(26,33)(28,34)(30,32);(3,5,7,4)(6,9,10,8)(11,13,15,17)(12,14,16,18)(19,22)(20,26)(21,30)(23,25)(24,32)(27,29)(28,31)(33,34);(3,6)(4,8)(5,9)(7,10)(11,14)(12,17)(13,16)(15,18)(19,23)(20,27)(21,31)(22,25)(24,33)(26,29)(28,30)(32,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010696);