Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#42246 |
| 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 | |
4486 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)(8,10)(11,12,15,16)(13,19,21,17)(14,18,22,23)(20,25,26,24)(27,28,32,30)(29,33,35,37)(31,34,39,38)(36,40,41,42);(3,4,7,6)(5,8,9,10)(12,16)(17,19)(18,23)(24,25)(29,35)(31,39)(33,37)(34,38);(3,5,7,9)(4,8,6,10)(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,23)(25,26)(27,29)(28,33)(30,37)(31,36)(32,35)(34,40)(38,42)(39,41);(11,13)(12,17)(14,20)(15,21)(16,19)(18,24)(22,26)(23,25)(27,30,32,28)(29,33,35,37)(31,38,39,34)(36,40,41,42);(3,6,7,4)(5,10,9,8)(11,14,15,22)(12,18,16,23)(13,20,21,26)(17,24,19,25)(27,31)(28,34)(29,36)(30,38)(32,39)(33,40)(35,41)(37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547807);