Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#49328 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5266 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)(5,9)(6,10)(7,8)(11,12,15,14)(13,17,18,16)(19,20,24,27)(21,22,25,29)(23,26,31,33)(28,30,32,34);(3,5)(4,6)(7,10)(8,9)(11,12,15,14)(13,16,18,17)(21,25)(22,29)(23,31)(26,33);(4,8)(5,10)(11,13,15,18)(12,16,14,17)(19,21)(20,22)(23,28)(24,25)(26,30)(27,29)(31,32)(33,34);(3,5)(4,9)(6,8)(7,10)(19,22)(20,25)(21,27)(23,30)(24,29)(26,32)(28,33)(31,34);(3,6,7,9)(4,5,8,10)(11,14,15,12)(13,17,18,16)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611469);