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×#55768 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4442 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)(5,9)(6,8)(7,10)(11,12,15,18)(13,19,21,16)(14,17,22,24)(20,25,26,23)(27,28)(29,35)(30,32)(31,38)(33,36)(34,39)(37,40)(41,42);(3,4,6,8)(5,7,10,9)(14,22)(17,24)(20,26)(23,25)(28,30)(29,36)(34,38)(37,42);(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26)(27,29)(28,33)(30,35)(31,37)(32,36)(34,40)(38,41)(39,42);(3,4)(5,7)(6,8)(9,10)(11,13,15,21)(12,16,18,19)(14,20,22,26)(17,23,24,25)(27,30,32,28)(29,35,36,33)(31,38,39,34)(37,41,42,40);(3,5,6,10)(4,7,8,9)(11,14)(12,17)(13,20)(15,22)(16,23)(18,24)(19,25)(21,26)(27,31,32,39)(28,34,30,38)(29,37,36,42)(33,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010162);