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 |
| direct product | C2×#22249 |
| 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×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1914 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)(25,31)(26,28)(27,35)(29,34)(30,33)(32,37)(36,38);(3,5,6,4)(7,9)(8,13)(10,19)(11,17)(12,18)(14,22)(15,16)(20,21)(24,28)(25,32)(30,35)(34,38);(7,10,12,11)(8,14,15,20)(9,17,18,19)(13,21,16,22)(23,25,33,38,26,32,27,34)(24,29,35,37,28,36,30,31);(7,11,12,10)(8,14,15,20)(9,17,18,19)(13,22,16,21)(24,28)(29,36)(30,35)(31,37)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283545);