Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33583 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2086 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)(9,19,22,36)(10,17,12,15)(11,24,27,32)(14,29,31,21)(16,23,34,26)(20,37,28,33)(25,35,38,30);(3,4,5,6);(7,9)(8,14)(10,20)(11,25)(12,28)(13,22)(15,30)(16,33)(17,35)(18,31)(19,24)(21,26)(23,29)(27,38)(32,36)(34,37);(7,10,13,12)(8,15,18,17)(9,20,22,28)(11,23,27,26)(14,30,31,35)(16,32,34,24)(19,33,36,37)(21,25,29,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530121);