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×#54456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4788 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);(3,4,7,5)(6,9,10,8)(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28)(29,33)(30,37)(31,34)(32,35)(36,40)(38,42)(39,41);(4,5)(8,9)(12,16)(13,19)(14,23)(24,26)(27,29,32,37)(28,33,35,30)(31,36,39,42)(34,40,41,38);(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29)(28,30)(31,36)(32,37)(33,35)(34,38)(39,42)(40,41);(3,5,7,4)(6,9,10,8)(11,13,15,19)(12,17,16,20)(14,21,23,25)(18,24,22,26)(27,30)(28,29)(31,38)(32,33)(34,36)(35,37)(39,40)(41,42);(3,6)(4,8)(5,9)(7,10)(11,14)(12,18)(13,21)(15,23)(16,22)(17,24)(19,25)(20,26)(27,31,32,39)(28,34,35,41)(29,36,37,42)(30,38,33,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010116);