Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#28911 |
| 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 | |
2478 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)(15,16,21,26)(17,23,28,24)(18,29,19,22)(20,30,27,25);(3,4,5,6)(7,9,11,10)(8,12,13,14)(15,17,27,18,21,28,20,19)(16,22,30,24,26,29,25,23);(4,6)(7,10,11,9)(8,12,13,14)(15,18,20,17,21,19,27,28)(16,23,25,29,26,24,30,22);(8,13)(12,14)(15,19,20,28,21,18,27,17)(16,24,25,22,26,23,30,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367153);