Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#42649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7264 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)(7,8,11,9)(10,13,14,12)(15,16,20,23)(17,21,24,19)(18,22,28,30)(25,29,26,27);(8,9)(12,13)(17,24)(19,21)(25,26)(27,29);(3,4)(5,6)(7,9,11,8)(10,13,14,12)(15,17)(16,21)(18,26)(19,23)(20,24)(22,27)(25,28)(29,30);(7,10,11,14)(8,12,9,13)(15,18)(16,22)(17,25)(19,27)(20,28)(21,29)(23,30)(24,26);(15,19,20,21)(16,17,23,24)(18,27,28,29)(22,25,30,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548472);