Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#27277 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2858 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)(15,16,21,26)(17,27,29,23)(18,30,28,22)(19,25,20,24);(3,4)(5,6)(8,11)(12,14)(15,17,20,18,21,29,19,28)(16,22,25,23,26,30,24,27);(4,6)(7,9,10,13)(8,12,11,14)(15,18,20,29,21,28,19,17)(16,23,25,30,26,27,24,22);(8,11)(12,14)(15,17,19,28,21,29,20,18)(16,22,24,27,26,30,25,23)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347939);