Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#46420 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2430 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8226 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)(15,16,20,21)(17,22,25,28)(18,23,27,30)(19,26,29,24);(7,8)(9,13)(10,11)(12,14)(16,21)(17,25)(18,27)(19,29);(3,4,5,6)(8,11)(12,13)(15,17)(16,22)(18,24)(19,23)(20,25)(21,28)(26,27)(29,30);(7,8)(9,12)(10,11)(13,14)(15,18)(16,23)(17,26)(19,28)(20,27)(21,30)(22,29)(24,25);(7,9)(8,12)(10,14)(11,13)(15,19,20,29)(16,24,21,26)(17,23,25,30)(18,28,27,22)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7607651);