Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9303 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1174 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);(4,8)(5,6)(9,10)(11,12)(13,17,22,32)(14,24)(15,27)(16,19)(18,25)(20,29)(21,35,36,42)(23,39,38,31)(26,34)(28,33)(30,40,41,37);(3,4,6,8,7,10,5,9)(11,13)(12,17,19,32)(14,21)(15,23)(16,22)(18,30,33,41)(20,31,27,39)(24,35,34,42)(25,37)(26,36)(28,40)(29,38);(3,5,7,6)(4,9,10,8)(11,14,16,26)(12,18,19,33)(13,21,22,36)(15,28,29,25)(17,30,32,41)(20,34,27,24)(23,40,38,37)(31,42,39,35)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106362);