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×#58158 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6168 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)(5,9)(6,7)(8,10)(11,12)(13,18)(14,17)(15,16)(19,20,24,28)(21,26,22,25)(23,31,32,27)(29,33,30,34);(4,7)(8,9)(11,13)(12,17)(14,16)(15,18)(19,21,24,22)(20,25,28,26)(23,30,32,29)(27,34,31,33);(13,14)(17,18)(19,22,24,21)(20,26,28,25)(23,30,32,29)(27,34,31,33);(3,4)(5,8)(6,7)(9,10)(11,14,16,13)(12,17,15,18)(19,20)(21,26)(22,25)(23,27)(24,28)(29,34)(30,33)(31,32);(3,5,6,10)(4,8,7,9)(11,15,16,12)(13,17,14,18)(19,23)(20,27)(21,29)(22,30)(24,32)(25,33)(26,34)(28,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011761);