Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52783 |
| 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 | |
4632 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);(3,4,7,5)(6,9,10,8)(11,12)(13,14)(15,23)(16,17)(18,21)(19,24)(20,26)(22,25)(30,38)(33,40)(36,42)(39,41);(3,4)(5,7)(6,8)(9,10)(12,17)(14,21)(19,23)(22,26)(27,28,31,34)(29,32,37,35)(30,33,38,40)(36,39,42,41);(4,5)(8,9)(11,13)(12,14)(15,20)(16,18)(17,21)(19,22)(23,26)(24,25)(27,28)(29,35)(30,33)(31,34)(32,37)(36,41)(38,40)(39,42);(3,5,7,4)(6,9,10,8)(11,14)(12,18)(13,17)(15,22)(16,21)(19,25)(20,23)(24,26)(27,29)(28,32)(30,36)(31,37)(33,39)(34,35)(38,42)(40,41);(3,6,7,10)(4,8,5,9)(11,15,16,24)(12,19,17,23)(13,20,18,25)(14,22,21,26)(27,30)(28,33)(29,36)(31,38)(32,39)(34,40)(35,41)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999724);