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×#21827 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,13)(6,17,20,10)(7,11)(8,12)(9,16)(14,28,29,25)(15,23)(18,32,33,22)(19,31,34,26)(21,27)(24,30);(3,5,7,16)(4,9,11,13)(6,18,19,14)(8,15,21,30)(10,25,26,22)(12,24,27,23)(17,28,31,32)(20,33,34,29);(3,6,7,19)(4,10,11,26)(5,14,16,18)(8,20,21,34)(9,22,13,25)(12,17,27,31)(15,29,30,33)(23,28,24,32);(4,11)(5,15)(9,23)(10,26)(12,27)(13,24)(14,29)(16,30)(17,31)(18,33)(22,28)(25,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281534);