Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17406 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,11,13,8)(6,9)(7,10)(12,17,18,15)(14,16)(19,20,22,26,23,25,29,32)(21,24,28,31,27,30,33,34);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,21)(20,24)(22,28)(23,27)(25,30)(26,31)(29,33)(32,34);(3,6)(4,9)(5,12)(7,14)(8,15)(10,16)(11,17)(13,18)(21,27)(24,30)(28,33)(31,34);(6,14)(9,16)(12,18)(15,17)(20,25)(24,30)(26,32)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249844);