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×#50334 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4714 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)(13,17)(14,21)(15,16)(18,22)(19,25)(20,23)(24,26)(27,28,32,33)(29,30,34,38)(31,39,40,35)(36,42,41,37);(3,4,7,6)(5,8,10,9)(12,16)(17,23)(18,21)(24,25)(28,33)(29,34)(31,40)(37,42);(3,5,7,10)(4,8,6,9)(11,13,15,20)(12,17,16,23)(14,19,22,26)(18,24,21,25)(27,29)(28,30)(31,36)(32,34)(33,38)(35,37)(39,42)(40,41);(3,6,7,4)(5,9,10,8)(11,12)(13,17)(14,18)(15,16)(19,24)(20,23)(21,22)(25,26)(27,30)(28,34)(29,33)(31,37)(32,38)(35,41)(36,39)(40,42);(11,14,15,22)(12,18,16,21)(13,19,20,26)(17,24,23,25)(27,31,32,40)(28,35,33,39)(29,36,34,41)(30,37,38,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613160);