Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37264 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4816 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);(4,7)(8,10)(11,12,15,16)(13,18,17,14)(19,20)(21,26)(22,25)(23,27)(24,29)(28,33)(30,32)(31,34);(12,16)(13,17)(19,21,25,24)(20,26,22,29)(23,31,32,28)(27,34,30,33);(3,4)(5,8)(6,7)(9,10)(11,13,15,17)(12,14,16,18)(19,22,25,20)(21,26,24,29)(23,30,32,27)(28,33,31,34);(11,14)(12,17)(13,16)(15,18)(19,23,25,32)(20,27,22,30)(21,28,24,31)(26,33,29,34);(3,5,6,9)(4,8,7,10)(19,24,25,21)(20,26,22,29)(23,31,32,28)(27,33,30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541818);