Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#8429 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3497 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3258 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);(3,4,6,8,7,10,5,9)(11,12,16,22,17,20,15,21)(13,23,25,36,26,18,31,33)(14,27,29,19,30,35,28,34)(24,38,39,32,40,42,41,37);(4,8)(5,6)(9,10)(11,13)(12,18)(14,24)(15,25)(16,31)(17,26)(19,32)(20,23)(21,33)(22,36)(27,42)(28,39)(29,41)(30,40)(34,37)(35,38);(11,14,17,30)(12,19,20,34)(13,24,26,40)(15,29,16,28)(18,32,23,37)(21,35,22,27)(25,41,31,39)(33,38,36,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93572);