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×#43493 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7696 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)(8,11)(12,13)(15,16,20,22)(17,19,23,28)(18,21,27,30)(24,26,25,29);(7,8,10,11)(9,13,14,12)(17,23)(19,28)(24,25)(26,29);(3,4)(5,6)(15,17)(16,19)(18,25)(20,23)(21,29)(22,28)(24,27)(26,30);(7,9,10,14)(8,12,11,13)(15,18)(16,21)(17,24)(19,26)(20,27)(22,30)(23,25)(28,29);(15,19)(16,17)(18,26)(20,28)(21,24)(22,23)(25,30)(27,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574874);