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×#44663 |
| 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 | |
7206 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)(9,13)(12,14)(15,16,19,20)(17,21,23,24)(18,26,27,22)(25,30,29,28);(3,4,5,6)(7,8,11,10)(9,12,13,14)(16,20)(17,23)(18,27)(28,30);(7,9,11,13)(8,12,10,14)(15,16)(17,24)(18,22)(19,20)(21,23)(25,30)(26,27)(28,29);(3,4)(5,6)(15,17)(16,21)(18,25)(19,23)(20,24)(22,28)(26,30)(27,29);(7,10,11,8)(9,14,13,12)(15,18,19,27)(16,22,20,26)(17,25,23,29)(21,28,24,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7575980);