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×#50342 |
| 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);(3,4,7,6)(5,9,10,8)(11,12)(13,18)(14,19)(15,23)(16,17)(20,24)(21,22)(25,26)(27,28,32,35)(29,31,34,40)(30,33,37,41)(36,39,38,42);(3,4,7,6)(5,8,10,9)(12,17)(14,18)(20,23)(22,25)(29,34)(30,37)(31,40)(33,41);(3,5,7,10)(4,8,6,9)(11,13,16,19)(12,18,17,14)(15,21,24,26)(20,25,23,22)(27,29,32,34)(28,31,35,40)(30,38,37,36)(33,42,41,39);(3,6,7,4)(5,9,10,8)(11,14,16,18)(12,19,17,13)(15,22,24,25)(20,26,23,21)(27,30)(28,33)(29,36)(31,39)(32,37)(34,38)(35,41)(40,42);(11,15,16,24)(12,20,17,23)(13,21,19,26)(14,22,18,25)(27,31)(28,34)(29,35)(30,39)(32,40)(33,38)(36,41)(37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613311);