Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53131 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(5,9)(7,10)(11,12)(13,14)(15,23)(16,17)(18,21)(19,24)(20,26)(22,25)(27,28,32,35)(29,33,30,37)(31,39,40,34)(36,42,38,41);(3,4)(5,7)(6,8)(9,10)(12,17)(14,21)(19,23)(22,26)(27,29,32,30)(28,33,35,37)(31,36,40,38)(34,41,39,42);(3,4,6,8)(5,7,9,10)(11,13)(12,14)(15,20)(16,18)(17,21)(19,22)(23,26)(24,25)(27,28)(29,33)(30,37)(31,34)(32,35)(36,41)(38,42)(39,40);(3,4)(5,7)(6,8)(9,10)(11,14)(12,18)(13,17)(15,22)(16,21)(19,25)(20,23)(24,26)(27,30,32,29)(28,33,35,37)(31,38,40,36)(34,41,39,42);(3,5,6,9)(4,7,8,10)(11,15,16,24)(12,19,17,23)(13,20,18,25)(14,22,21,26)(27,31,32,40)(28,34,35,39)(29,36,30,38)(33,41,37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000073);