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×#55713 |
| 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 | |
4474 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)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28,32,35)(29,33,30,37)(31,34,39,42)(36,40,38,41);(4,7)(8,10)(12,16)(13,19)(14,23)(24,26)(27,29,32,30)(28,33,35,37)(31,38,39,36)(34,41,42,40);(3,4,6,7)(5,8,9,10)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,30,32,29)(28,33,35,37)(31,38,39,36)(34,40,42,41);(3,5)(4,8)(6,9)(7,10)(11,13,15,19)(12,17,16,20)(14,21,23,25)(18,24,22,26)(27,28)(29,33)(30,37)(31,34)(32,35)(36,40)(38,41)(39,42);(11,14)(12,18)(13,21)(15,23)(16,22)(17,24)(19,25)(20,26)(27,31)(28,34)(29,36)(30,38)(32,39)(33,40)(35,42)(37,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006864);