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×#56904 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4902 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)(6,9)(8,10)(11,12,16,17)(13,15,14,18)(19,20,24,27)(21,28,25,22)(23,26,32,31)(29,34,33,30);(4,5)(6,10)(12,17)(13,14)(19,20,24,27)(21,22,25,28)(23,31,32,26)(29,34,33,30);(4,5)(8,9)(11,13,16,14)(12,18,17,15)(19,21)(20,22)(23,29)(24,25)(26,30)(27,28)(31,34)(32,33);(3,5,7,4)(6,9,10,8)(11,14,16,13)(12,15,17,18)(19,22)(20,25)(21,27)(23,30)(24,28)(26,33)(29,31)(32,34);(3,6)(4,8)(5,9)(7,10)(11,15)(12,13)(14,17)(16,18)(19,23,24,32)(20,26,27,31)(21,29,25,33)(22,30,28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006823);