Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9429 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1412 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);(3,4,8,10,7,5,6,9)(11,12,16,20,17,22,15,21)(13,23,26,33,27,36,25,18)(14,28,30,19,31,34,29,35)(24,32,39,41,40,42,38,37);(3,5,7,4)(6,9,8,10)(11,13)(12,18,22,33)(14,24)(15,25)(16,26)(17,27)(19,32,35,42)(20,23,21,36)(28,37,34,41)(29,38)(30,39)(31,40);(6,8)(9,10)(11,14,17,31)(12,19,22,35)(13,24,27,40)(15,30,16,29)(18,32,33,42)(20,28,21,34)(23,37,36,41)(25,39,26,38)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108036);