Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26833 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2298 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)(9,14)(10,12)(11,13)(15,16)(17,23)(18,22)(19,26)(20,25)(21,27)(24,29)(28,30);(3,4)(5,6)(7,9,11,10)(8,12,13,14)(15,17,19,18,21,29,20,28)(16,22,25,23,27,30,26,24);(4,6)(7,10,11,9)(8,12,13,14)(15,18,19,29,21,28,20,17)(16,23,25,30,27,24,26,22);(8,13)(12,14)(15,18,19,29,21,28,20,17)(16,24,25,22,27,23,26,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351061);