Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18329 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#538 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3558 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);(3,4,8,13)(5,15,18,24)(6,11,7,21)(9,14,19,12)(10,22,27,16)(17,31,32,30)(20,29,23,28)(25,33,26,34);(3,5)(4,10)(6,16)(7,22)(8,18)(9,17)(11,24)(12,25)(13,27)(14,26)(15,21)(19,32)(20,33)(23,34)(28,31)(29,30);(3,6,19,23)(4,11,12,28)(5,16,32,34)(7,9,20,8)(10,24,25,31)(13,21,14,29)(15,26,30,27)(17,33,18,22);(4,12)(10,25)(11,28)(13,14)(15,30)(21,29)(24,31)(26,27)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254072);