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×#58942 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,8,6)(5,9,7,10)(12,14)(13,15);(4,6)(5,7)(11,12)(13,17)(14,16)(15,18)(19,20,23,26)(21,24,29,27)(22,30,31,25)(28,33,34,32);(3,5,8,7)(4,9,6,10)(11,13,16,15)(12,17,14,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,31)(32,34);(3,6,8,4)(5,9,7,10)(11,14,16,12)(13,17,15,18)(19,21)(20,24)(22,28)(23,29)(25,32)(26,27)(30,33)(31,34);(3,7,8,5)(4,9,6,10)(11,15,16,13)(12,17,14,18)(19,22,23,31)(20,25,26,30)(21,28,29,34)(24,32,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012602);