Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53219 |
| 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 | |
4842 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)(5,6)(7,8)(9,10)(13,17)(16,18)(19,20,25,28)(21,23,26,22)(24,32,33,27)(29,30,34,31);(4,8)(5,9)(12,15)(16,18)(19,21,25,26)(20,22,28,23)(24,29,33,34)(27,30,32,31);(11,12)(13,16)(14,15)(17,18)(19,22,25,23)(20,26,28,21)(24,30,33,31)(27,34,32,29);(3,5,7,9)(4,6,8,10)(11,13)(12,16)(14,17)(15,18)(19,23,25,22)(20,21,28,26)(24,31,33,30)(27,29,32,34);(3,6)(4,9)(5,8)(7,10)(11,12)(13,16)(14,15)(17,18)(19,24)(20,27)(21,29)(22,30)(23,31)(25,33)(26,34)(28,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012868);