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×#56998 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4750 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);(3,4,7,9)(5,6,10,8)(11,12,16,14)(13,17,19,21)(15,22,23,18)(20,26,25,24)(27,28)(29,34)(30,35)(31,39)(32,33)(36,40)(37,38)(41,42);(3,5,7,10)(4,8,9,6)(12,14)(13,19)(15,23)(24,26)(28,33)(30,34)(36,39)(38,41);(4,9)(6,8)(11,13)(12,17)(14,21)(15,20)(16,19)(18,24)(22,26)(23,25)(27,29,32,35)(28,34,33,30)(31,37,40,42)(36,41,39,38);(3,6,7,8)(4,5,9,10)(11,14,16,12)(13,17,19,21)(15,22,23,18)(20,24,25,26)(27,30,32,34)(28,35,33,29)(31,38,40,41)(36,42,39,37);(3,5)(4,8)(6,9)(7,10)(11,15,16,23)(12,18,14,22)(13,20,19,25)(17,24,21,26)(27,31,32,40)(28,36,33,39)(29,37,35,42)(30,38,34,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006857);