Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#29875 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1860 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);(3,4,5,6)(7,8)(9,13)(10,14)(11,12)(15,21)(16,19)(17,22)(18,20)(23,24)(25,32)(26,36)(27,31)(28,30)(29,35)(33,38)(34,37);(8,12)(13,19)(14,20)(21,22)(23,25,27,35)(24,29,31,32)(26,34,28,33)(30,37,36,38);(4,6)(7,9,11,16)(8,13,12,19)(10,17,18,15)(14,22,20,21)(25,33)(26,28)(29,37)(30,36)(32,38)(34,35);(7,10,11,18)(8,14,12,20)(9,15,16,17)(13,21,19,22)(23,26,27,28)(24,30,31,36)(25,34,35,33)(29,37,32,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343344);