Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#34488 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(3,4,6,5)(7,8)(9,13)(10,12)(11,14)(15,16)(17,20)(18,23)(19,24)(21,25)(22,27)(26,28)(29,30);(7,8,10,12)(9,11,14,13)(16,17)(18,25)(22,24)(26,30);(15,17,20,16)(18,23,25,21)(19,24,27,22)(26,29,30,28);(3,5,6,4)(7,8)(9,11)(10,12)(13,14)(15,18)(16,21)(17,23)(19,26)(20,25)(22,28)(24,29)(27,30);(7,9,10,14)(8,11,12,13)(15,19,20,27)(16,22,17,24)(18,26,25,30)(21,28,23,29)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533325);