Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(7,8)(9,14)(10,12)(11,13)(15,16,19,24,20,22,18,23)(17,25,26,29,28,21,27,30);(3,4)(5,6)(7,9,11,10)(8,12,13,14)(15,17,20,28)(16,21,22,25)(18,26,19,27)(23,30,24,29);(4,6)(7,10,11,9)(8,12,13,14)(16,22)(17,26)(18,19)(21,29)(25,30)(27,28)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265786);