Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#12851 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1817 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,11,7,12,18,25,8,9,17,20,19,21,30,31)(5,13,15,26,16,27,29,34,14,10,22,23,28,24,32,33);(4,9)(6,17)(10,13)(12,21)(15,22)(18,30)(24,27)(29,32);(3,5)(4,10,9,13)(6,15)(7,16)(8,14)(11,23,20,26)(12,24,21,27)(17,22)(18,29)(19,28)(25,33,31,34)(30,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401043);