Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#17209 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,6)(7,8,11,9)(10,13,14,12)(15,16)(17,20)(18,19)(21,22)(23,24)(25,29)(26,30)(27,28);(8,9)(10,14)(16,18)(20,22)(23,25,28,26)(24,29,27,30);(7,9,11,8)(10,12,14,13)(15,17,19,21)(16,20,18,22)(23,26,28,25)(24,29,27,30);(3,4)(5,6)(7,10,11,14)(8,12,9,13)(15,18,19,16)(17,20,21,22)(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, 752192);