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×#56392 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2431 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5906 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)(5,7)(6,8)(9,10)(19,20,24,27)(21,22,29,25)(23,31,32,26)(28,33,34,30);(6,10)(8,9)(11,12)(13,17)(14,16)(15,18)(19,21,24,29)(20,25,27,22)(23,28,32,34)(26,33,31,30);(4,5)(8,9)(11,12,14,16)(13,15,18,17)(19,22,24,25)(20,21,27,29)(23,30,32,33)(26,28,31,34);(3,5,7,4)(6,9,10,8)(11,12)(13,15)(14,16)(17,18)(19,21)(20,25)(22,27)(23,28)(24,29)(26,33)(30,31)(32,34);(3,6)(4,8)(5,9)(7,10)(11,13)(12,15)(14,18)(16,17)(19,23,24,32)(20,26,27,31)(21,28,29,34)(22,30,25,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006006);