Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#42788 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(8,11)(9,13)(15,16)(17,19)(18,26)(20,21)(22,25)(23,29)(24,30)(27,28);(7,8,10,11)(9,12,13,14)(16,21)(19,22)(23,26)(28,30);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,17)(16,22)(18,27)(19,21)(20,25)(23,28)(24,29)(26,30);(7,9,10,13)(8,12,11,14)(15,18,20,29)(16,23,21,26)(17,24,25,27)(19,28,22,30);(15,19,20,22)(16,17,21,25)(18,28,29,30)(23,24,26,27)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548608);