Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56999 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,10)(6,9)(7,8)(11,12,16,14)(13,17,21,19)(15,22,23,18)(20,25,26,24)(27,28,32,35)(29,33,30,37)(31,34,39,42)(36,40,38,41);(3,5)(4,9)(6,8)(7,10)(12,14)(17,19)(18,22)(24,25)(27,29,32,30)(28,33,35,37)(31,38,39,36)(34,41,42,40);(5,6)(9,10)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,30,32,29)(28,33,35,37)(31,38,39,36)(34,40,42,41);(3,6,8,5)(4,9,7,10)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,28)(29,33)(30,37)(31,34)(32,35)(36,40)(38,41)(39,42);(3,7,8,4)(5,9,6,10)(11,15,16,23)(12,18,14,22)(13,20,21,26)(17,24,19,25)(27,31,32,39)(28,34,35,42)(29,36,30,38)(33,40,37,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006889);