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×#52635 |
| 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 | |
5070 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);(3,4,6,7)(5,9,10,8)(13,17)(16,18)(19,20,24,25)(21,28,31,26)(22,27,32,23)(29,30,34,33);(11,12)(13,16)(14,15)(17,18)(20,25)(22,32)(26,28)(29,34);(4,7)(8,9)(12,15)(16,18)(19,21,24,31)(20,26,25,28)(22,29,32,34)(23,30,27,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,22)(20,27)(21,29)(23,25)(24,32)(26,33)(28,30)(31,34);(3,5,6,10)(4,8,7,9)(11,13)(12,16)(14,17)(15,18)(19,23,24,27)(20,22,25,32)(21,30,31,33)(26,29,28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999342);