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×#54809 |
| 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 | |
4718 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);(3,4,9,10)(5,6,7,8)(11,12,15,16)(13,19,20,17)(14,22,23,18)(21,24,26,25)(27,28)(29,35)(30,37)(31,39)(32,33)(34,40)(36,38)(41,42);(3,5,9,7)(4,6,10,8)(12,16)(13,20)(18,22)(21,26)(28,33)(30,35)(31,40)(36,42);(3,6,9,8)(4,5,10,7)(11,13,15,20)(12,17,16,19)(14,21,23,26)(18,24,22,25)(27,29,32,37)(28,30,33,35)(31,36,40,42)(34,38,39,41);(3,7,9,5)(4,6,10,8)(11,12)(13,17)(14,18)(15,16)(19,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,38)(32,35)(33,37)(34,36)(39,42)(40,41);(3,8,9,6)(4,5,10,7)(11,14)(12,18)(13,21)(15,23)(16,22)(17,24)(19,25)(20,26)(27,31)(28,34)(29,36)(30,38)(32,40)(33,39)(35,41)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008649);