Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#17214 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,6)(7,8,10,12)(9,11,13,14)(15,16)(17,18)(19,20)(21,22)(23,24,28,27)(25,29,26,30);(9,13)(11,14)(16,20)(17,22)(23,25,28,26)(24,29,27,30);(7,9)(8,11)(10,13)(12,14)(15,17)(16,21)(18,20)(19,22)(23,26,28,25)(24,29,27,30);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,18,19,21)(16,22,20,17)(23,27,28,24)(25,29,26,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752198);