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×#42230 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4594 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);(3,4,7,10)(5,8,6,9)(11,12)(13,16)(14,19)(15,18)(17,22)(20,23)(21,25)(24,26)(27,28,32,30)(29,33,35,37)(31,34,39,38)(36,40,41,42);(3,5,7,6)(4,8,10,9)(12,13)(17,19)(18,20)(24,25)(29,35)(31,39)(33,37)(34,38);(3,4)(5,8)(6,9)(7,10)(11,13,16,12)(14,19,22,17)(15,20,23,18)(21,25,26,24)(27,29)(28,33)(30,37)(31,36)(32,35)(34,40)(38,42)(39,41);(11,14)(12,17)(13,19)(15,21)(16,22)(18,24)(20,25)(23,26)(27,30,32,28)(29,33,35,37)(31,38,39,34)(36,40,41,42);(3,6,7,5)(4,9,10,8)(11,15,16,23)(12,18,13,20)(14,21,22,26)(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, 7547806);