Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#15232 |
| 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 | |
3446 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);(7,8,10,11)(9,13,14,12)(17,21)(19,22)(24,28)(26,29);(4,6)(8,11)(12,13)(15,16,18,20)(17,19,21,22)(23,24)(25,29)(26,30)(27,28);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22)(23,25,27,30)(24,29,28,26);(7,9,10,14)(8,12,11,13)(15,17,18,21)(16,19,20,22)(23,26,27,29)(24,30,28,25)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515750);