Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#35637 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7456 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)(8,9)(13,14)(15,16,20,18)(17,21,23,25)(19,22,27,26)(24,28,29,30);(7,8,12,9)(10,13,11,14)(17,23)(19,27)(21,25)(22,26);(3,4,5,6)(7,9,12,8)(10,14,11,13)(15,17)(16,21)(18,25)(19,24)(20,23)(22,28)(26,30)(27,29);(7,10,12,11)(8,13,9,14)(15,18,20,16)(17,21,23,25)(19,26,27,22)(24,28,29,30);(7,11,12,10)(8,14,9,13)(15,19)(16,22)(17,24)(18,26)(20,27)(21,28)(23,29)(25,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536238);