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×#42481 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8322 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)(15,16,21,26)(17,23,18,22)(19,20,25,24)(27,28,29,30);(7,8)(9,13)(10,11)(12,14)(15,17,21,18)(16,22,26,23)(19,27,25,29)(20,30,24,28);(3,4,5,6)(8,11)(12,13)(15,18,21,17)(16,23,26,22)(19,27,25,29)(20,30,24,28);(7,8)(9,12)(10,11)(13,14)(15,19,21,25)(16,24,26,20)(17,27,18,29)(22,28,23,30);(7,9,10,14)(8,12,11,13)(15,20,21,24)(16,25,26,19)(17,28,18,30)(22,27,23,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548309);