Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8346 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8348 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#534 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2328 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)(11,12,15,20,16,19,28,33)(13,21,24,34,23,17,27,30)(14,18,26,31,25,32,38,42)(22,29,37,40,36,35,39,41);(3,4)(5,9)(6,8)(7,10)(11,13)(12,17)(14,22)(15,27)(16,23)(18,29)(19,21)(20,34)(24,28)(25,36)(26,39)(30,33)(31,41)(32,35)(37,38)(40,42);(11,14)(12,18)(13,22)(15,26)(16,25)(17,29)(19,32)(20,31)(21,35)(23,36)(24,37)(27,39)(28,38)(30,40)(33,42)(34,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92487);