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×#57863 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4680 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)(5,7)(6,8)(9,10)(11,12)(13,16)(14,19)(15,20)(17,21)(18,23)(22,24)(25,26)(27,28,32,35)(29,30,37,33)(31,39,41,34)(36,42,40,38);(6,10)(8,9)(12,13)(14,21)(18,20)(22,26)(27,29,32,37)(28,33,35,30)(31,40,41,36)(34,38,39,42);(4,5)(8,9)(11,13,16,12)(14,19,21,17)(15,20,23,18)(22,25,26,24)(27,30)(28,29)(31,38)(32,33)(34,36)(35,37)(39,40)(41,42);(3,5,7,4)(6,9,10,8)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,29)(28,33)(30,35)(31,36)(32,37)(34,42)(38,39)(40,41);(3,6,7,10)(4,8,5,9)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,31)(28,34)(29,36)(30,38)(32,41)(33,42)(35,39)(37,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008553);