Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8504 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #379 |
| prime radical | C2×#3529 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,7,8,9)(11,12)(13,21,24,17)(14,25,26,36)(15,20)(16,19)(18,27,33,29)(22,38)(23,35,28,32)(30,34)(31,40)(37,41)(39,42);(3,4)(5,9)(6,8)(7,10)(11,13)(12,17)(14,22)(15,28)(16,24)(18,31)(19,21)(20,35)(23,30)(25,38)(26,39)(27,41)(29,40)(32,34)(33,37)(36,42);(5,10)(7,9)(11,14)(12,18)(13,22)(15,29)(16,26)(17,31)(19,33)(20,36)(21,37)(23,41)(24,39)(25,34)(27,30)(28,40)(32,38)(35,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94772);