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×#53251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4652 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);(5,9)(8,10)(11,12,15,18)(13,16,20,23)(14,17,21,24)(19,22,25,26)(27,28,32,30)(29,35,37,33)(31,38,39,34)(36,40,42,41);(4,7)(8,10)(14,21)(17,24)(19,25)(22,26)(28,30)(31,39)(33,35)(36,42);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,26)(27,29,32,37)(28,33,30,35)(31,36,39,42)(34,40,38,41);(3,5)(4,8)(6,9)(7,10)(11,13,15,20)(12,16,18,23)(14,19,21,25)(17,22,24,26)(27,30,32,28)(29,35,37,33)(31,38,39,34)(36,41,42,40);(3,4)(5,8)(6,7)(9,10)(11,14)(12,17)(13,19)(15,21)(16,22)(18,24)(20,25)(23,26)(27,31,32,39)(28,34,30,38)(29,36,37,42)(33,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012891);