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×#58415 |
| 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 | |
5442 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);(3,4,6,7)(5,8,9,10)(11,12,14,16)(13,17,18,15)(19,20)(21,26)(22,28)(23,25)(24,27)(29,33)(30,34)(31,32);(4,7)(5,9)(13,18)(15,17)(19,21,25,22)(20,26,23,28)(24,30,32,29)(27,34,31,33);(3,5,6,9)(4,8,7,10)(11,13,14,18)(12,15,16,17)(19,22,25,21)(20,26,23,28)(24,30,32,29)(27,33,31,34);(3,5)(4,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,23,25,20)(21,28,22,26)(24,31,32,27)(29,34,30,33);(3,4)(5,8)(6,7)(9,10)(19,24)(20,27)(21,29)(22,30)(23,31)(25,32)(26,33)(28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011473);