Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#22253 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,6,5)(7,8)(9,16)(10,14)(11,20)(12,15)(13,18)(17,22)(19,21)(23,24)(25,32)(26,30)(27,36)(28,31)(29,34)(33,37)(35,38);(3,5,6,4)(7,9)(8,13)(10,19)(11,17)(12,18)(14,22)(15,16)(20,21)(23,24)(25,29)(26,36)(27,30)(28,31)(32,34)(33,38)(35,37);(7,10,12,11)(8,14,15,20)(9,17,18,19)(13,21,16,22)(23,25,27,34,28,35,26,33)(24,29,30,32,31,37,36,38);(7,11,12,10)(8,14,15,20)(9,17,18,19)(13,22,16,21)(23,26,28,27)(24,30,31,36)(25,33,35,34)(29,32,37,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283548);