Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C43 |
| direct product | C2×#30273 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(7,8)(9,12)(10,11)(13,14)(15,16,21,27,20,18,19,23)(17,22,28,30,25,24,26,29);(3,4,5,6)(8,11)(12,14)(15,17)(16,22)(18,24)(19,26)(20,25)(21,28)(23,29)(27,30);(4,6)(9,13)(12,14)(15,18,20,16)(17,24,25,22)(19,23,21,27)(26,29,28,30);(7,9,10,13)(8,12,11,14)(17,25)(22,24)(26,28)(29,30)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343134);