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×#47684 |
| 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 | |
7588 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)(15,16,20,24)(17,18,22,21)(19,27,29,23)(25,30,28,26);(7,8)(9,10)(11,12)(13,14)(15,17,20,22)(16,21,24,18)(19,28,29,25)(23,26,27,30);(3,4)(5,6)(8,12)(9,13)(15,17)(16,18)(19,25)(20,22)(21,24)(23,26)(27,30)(28,29);(7,9,11,13)(8,10,12,14)(15,18)(16,22)(17,24)(19,26)(20,21)(23,28)(25,27)(29,30);(7,10)(8,13)(9,12)(11,14)(15,19,20,29)(16,23,24,27)(17,25,22,28)(18,26,21,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608890);