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×#57373 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(12,16)(13,19)(18,23)(20,25)(27,28,32,30)(29,33,35,37)(31,38,39,34)(36,42,41,40);(3,4,7,9)(5,6,10,8)(12,16)(17,22)(18,23)(24,26)(28,30)(29,35)(31,39)(40,42);(3,5,7,10)(4,8,9,6)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,29)(28,33)(30,37)(31,36)(32,35)(34,40)(38,42)(39,41);(3,5)(4,6)(7,10)(8,9)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,30,32,28)(29,33,35,37)(31,38,39,34)(36,40,41,42);(3,6)(4,5)(7,8)(9,10)(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(27,31,32,39)(28,34,30,38)(29,36,35,41)(33,40,37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007924);