Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#42704 |
| 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 | |
5122 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);(3,4,7,8)(5,6,10,9)(12,14)(17,18)(19,20,24,27)(21,25,28,23)(22,26,32,34)(29,33,30,31);(4,8)(6,9)(11,12,16,14)(13,17,15,18)(21,28)(23,25)(29,30)(31,33);(4,8)(5,10)(11,13,16,15)(12,17,14,18)(19,21)(20,25)(22,30)(23,27)(24,28)(26,31)(29,32)(33,34);(3,5,7,10)(4,9,8,6)(11,14,16,12)(13,17,15,18)(19,22)(20,26)(21,29)(23,31)(24,32)(25,33)(27,34)(28,30);(3,6,7,9)(4,10,8,5)(11,15,16,13)(12,17,14,18)(19,23,24,25)(20,21,27,28)(22,31,32,33)(26,29,34,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548523);