Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54308 |
| 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 | |
4620 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);(4,7)(5,9)(11,12,16,14)(13,17,21,19)(15,22,23,18)(20,25,26,24)(27,28)(29,34)(30,35)(31,38)(32,33)(36,41)(37,40)(39,42);(3,4,6,7)(5,8,9,10)(12,14)(17,19)(18,22)(24,25)(28,33)(29,37)(30,39)(31,41);(3,4)(5,8)(6,7)(9,10)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,29,32,37)(28,34,33,40)(30,36,39,38)(31,35,41,42);(3,4)(5,8)(6,7)(9,10)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,30,32,39)(28,35,33,42)(29,38,37,36)(31,40,41,34);(3,5)(4,8)(6,9)(7,10)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,31)(28,36)(29,35)(30,40)(32,41)(33,38)(34,39)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010699);