Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33590 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2062 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);(4,6)(7,8,13,18)(9,14,21,32)(10,16,12,28)(11,24,27,38)(15,33,35,20)(17,23,36,26)(19,29,22,31)(25,30,37,34);(3,4,5,6)(7,9)(8,14)(10,22)(11,20)(12,19)(13,21)(15,24)(16,31)(17,30)(18,32)(23,37)(25,26)(27,33)(28,29)(34,36)(35,38);(7,9)(8,15)(10,19)(11,25)(12,22)(13,21)(14,24)(16,30)(17,31)(18,35)(20,26)(23,33)(27,37)(28,34)(29,36)(32,38);(7,10,13,12)(8,16,18,28)(9,19,21,22)(11,23,27,26)(14,29,32,31)(15,30,35,34)(17,24,36,38)(20,25,33,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530129);