Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#30574 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2784 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);(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,24)(18,27)(19,21)(20,23)(22,26)(25,29)(28,30);(7,9)(8,12)(10,13)(11,14)(16,21)(17,25)(18,20)(22,29)(24,30)(26,28);(4,6)(7,10,11,12)(8,9,13,14)(15,17,19,26)(16,22,21,24)(18,25,20,28)(23,29,27,30);(4,6)(8,13)(9,14)(15,18,19,20)(16,23,21,27)(17,25,26,28)(22,29,24,30)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408370);