Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52068 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4448 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);(5,9)(7,10)(11,12,16,17)(13,19,20,14)(15,23,24,18)(21,22,26,25)(27,28,32,35)(29,33,36,31)(30,34,38,42)(37,41,39,40);(3,4,6,8)(5,7,9,10)(12,17)(13,20)(15,24)(22,25)(29,36)(30,38)(31,33)(34,42);(3,5,6,9)(4,7,8,10)(11,13,16,20)(12,14,17,19)(15,21,24,26)(18,22,23,25)(27,29)(28,33)(30,39)(31,35)(32,36)(34,40)(37,38)(41,42);(11,14)(12,13)(15,22)(16,19)(17,20)(18,21)(23,26)(24,25)(27,30,32,38)(28,34,35,42)(29,37,36,39)(31,40,33,41);(11,15,16,24)(12,18,17,23)(13,21,20,26)(14,22,19,25)(27,31,32,33)(28,29,35,36)(30,40,38,41)(34,37,42,39)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615327);