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×#52631 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(5,9)(8,10)(12,16)(13,19)(18,23)(20,25)(27,28,33,32)(29,35,30,34)(31,36,40,39)(37,42,38,41);(3,4)(5,8)(6,7)(9,10)(12,16)(17,22)(18,23)(24,26)(27,29,33,30)(28,34,32,35)(31,38,40,37)(36,42,39,41);(4,7)(8,10)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,23)(25,26)(27,30,33,29)(28,35,32,34)(31,38,40,37)(36,42,39,41);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,31,33,40)(28,36,32,39)(29,37,30,38)(34,41,35,42);(3,5)(4,8)(6,9)(7,10)(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(27,32,33,28)(29,35,30,34)(31,39,40,36)(37,42,38,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999436);