Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#45217 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7478 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);(3,4)(5,6)(8,9)(12,13)(15,16,20,24)(17,21,25,27)(18,22,28,30)(19,26,29,23);(4,6)(7,8,11,9)(10,13,14,12)(17,25)(19,29)(21,27)(23,26);(7,9,11,8)(10,13,14,12)(15,17)(16,21)(18,23)(19,22)(20,25)(24,27)(26,28)(29,30);(3,4)(5,6)(7,10,11,14)(8,12,9,13)(15,18)(16,22)(17,26)(19,27)(20,28)(21,29)(23,25)(24,30);(15,19,20,29)(16,23,24,26)(17,22,25,30)(18,27,28,21)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7576790);