Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9669 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #390 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,6,9,7,5,8,10)(11,12)(13,17,22,31)(14,24)(15,28)(16,19)(18,26)(20,25)(21,33,35,42)(23,37,34,30)(27,38)(29,40,41,36)(32,39);(3,5,6,10,7,4,8,9)(11,13,16,22)(12,17)(14,21,27,35)(15,23,25,34)(18,29)(19,31)(20,30)(24,33)(26,36,39,40)(28,37)(32,41)(38,42);(11,14,25,39,16,27,15,26)(12,18,28,38,19,32,20,24)(13,21,34,40,22,35,23,36)(17,29,37,42,31,41,30,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111624);