Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52396 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5984 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);(5,9)(8,10)(12,15)(13,17)(19,20,25,23)(21,28,26,22)(24,31,32,27)(29,30,33,34);(3,4)(5,8)(6,7)(9,10)(19,21,25,26)(20,22,23,28)(24,29,32,33)(27,30,31,34);(4,7)(8,10)(11,12,14,15)(13,16,17,18)(19,22,25,28)(20,26,23,21)(24,30,32,34)(27,33,31,29);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,23,25,20)(21,22,26,28)(24,31,32,27)(29,30,33,34);(3,5)(4,8)(6,9)(7,10)(11,13)(12,16)(14,17)(15,18)(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, 7999325);