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×#41369 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7760 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);(4,6)(7,8)(9,13)(10,11)(12,14)(15,16,20,22)(17,23,24,19)(18,21,28,30)(25,29,26,27);(8,11)(12,13)(17,24)(19,23)(25,26)(27,29);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,17)(16,19)(18,26)(20,24)(21,29)(22,23)(25,28)(27,30);(7,9,10,14)(8,12,11,13)(15,18)(16,21)(17,25)(19,27)(20,28)(22,30)(23,29)(24,26);(15,19)(16,17)(18,27)(20,23)(21,25)(22,24)(26,30)(28,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546934);