Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#36973 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(3,4,5,6)(7,8,10,11)(9,13,14,12)(18,25)(22,27)(24,29)(28,30);(8,11)(12,13)(16,20)(17,23)(22,27)(24,29);(3,4)(5,6)(15,16,19,20)(17,21,23,26)(18,22,25,27)(24,28,29,30);(7,8)(9,12)(10,11)(13,14)(15,17)(16,21)(18,24)(19,23)(20,26)(22,28)(25,29)(27,30);(7,9,10,14)(8,12,11,13)(15,18)(16,22)(17,24)(19,25)(20,27)(21,28)(23,29)(26,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541570);