Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#15027 |
| 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 | |
3382 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);(4,5)(8,11)(9,13)(15,16,18,20)(17,21,22,19)(23,24,27,28)(25,29,30,26);(3,4,6,5)(17,22)(19,21)(24,28)(25,30);(7,8,10,11)(9,12,13,14)(15,17,18,22)(16,19,20,21)(23,25)(24,29)(26,28)(27,30);(3,5,6,4)(7,9)(8,12)(10,13)(11,14)(15,16)(17,19)(18,20)(21,22)(23,26,27,29)(24,25,28,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515304);