Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53311 |
| 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 | |
4468 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)(13,19)(17,22)(20,25)(24,26)(27,28,32,36)(29,37,40,33)(30,34,35,31)(38,39,41,42);(4,7)(8,10)(12,16)(14,21)(17,22)(20,25)(27,29,32,40)(28,33,36,37)(30,41,35,38)(31,42,34,39);(3,4,6,7)(5,8,9,10)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,30,32,35)(28,34,36,31)(29,38,40,41)(33,42,37,39);(3,5,6,9)(4,8,7,10)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,29)(28,33)(30,38)(31,39)(32,40)(34,42)(35,41)(36,37);(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,31,32,34)(28,35,36,30)(29,39,40,42)(33,41,37,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013083);