Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#21828 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3174 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)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20)(21,27,29,24)(22,25)(23,26)(28,33,34,31)(30,32);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(22,30)(25,32)(28,34)(31,33);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,21)(20,24)(22,28)(23,29)(25,31)(26,27)(30,34)(32,33);(4,10)(8,11)(9,18)(16,17)(19,22,23,30)(20,25,26,32)(21,28,29,34)(24,31,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281535);