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×#57744 |
| 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 | |
4938 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);(3,4,7,9)(5,10,6,8)(12,16)(13,19)(18,23)(21,25)(28,29)(30,37)(31,39)(40,41);(3,5)(4,8)(6,7)(9,10)(14,22)(18,23)(21,25)(24,26)(27,28,32,29)(30,35,37,33)(31,36,39,34)(38,40,42,41);(5,6)(8,10)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29,32,28)(30,33,37,35)(31,36,39,34)(38,40,42,41);(3,6,7,5)(4,8,9,10)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,25)(23,26)(27,30,32,37)(28,33,29,35)(31,38,39,42)(34,40,36,41);(3,4)(5,10)(6,8)(7,9)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(27,31,32,39)(28,34,29,36)(30,38,37,42)(33,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008916);