Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#21837 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,5,6)(7,8)(9,15)(10,13)(11,14)(12,17)(16,21)(18,20)(19,22)(23,24)(25,33)(26,32)(27,29)(28,31)(30,35)(34,38)(36,37);(7,9,11,17)(8,12,14,15)(10,18,19,16)(13,21,22,20)(24,29)(25,34)(31,32)(35,36);(7,10,11,19)(8,13,14,22)(9,16,17,18)(12,20,15,21)(23,25,27,34)(24,30,29,37)(26,35,28,36)(31,33,32,38);(8,14)(12,15)(13,22)(20,21)(23,26,27,28)(24,31,29,32)(25,35,34,36)(30,33,37,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281553);