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×#57456 |
| 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 | |
4702 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);(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28,31,34)(29,35,36,32)(30,33,38,40)(37,41,42,39);(3,4,7,9)(5,6,10,8)(12,16)(13,19)(14,23)(24,26)(29,36)(32,35)(37,42)(39,41);(3,5,7,10)(4,8,9,6)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29)(28,32)(30,37)(31,36)(33,39)(34,35)(38,42)(40,41);(3,5)(4,6)(7,10)(8,9)(11,13,15,19)(12,17,16,20)(14,21,23,25)(18,24,22,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40)(41,42);(3,6)(4,5)(7,8)(9,10)(11,14)(12,18)(13,21)(15,23)(16,22)(17,24)(19,25)(20,26)(27,30,31,38)(28,33,34,40)(29,37,36,42)(32,39,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008191);