Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#50186 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4906 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);(5,9)(7,10)(11,12)(13,18)(14,19)(15,23)(16,17)(20,24)(21,22)(25,26)(27,28,31,32)(29,33,35,39)(30,34,38,40)(36,41,37,42);(3,4,6,8)(5,7,9,10)(12,17)(14,18)(20,23)(22,25)(28,32)(29,35)(34,40)(36,37);(3,5)(4,7)(6,9)(8,10)(11,13,16,19)(12,18,17,14)(15,21,24,26)(20,25,23,22)(27,29)(28,33)(30,37)(31,35)(32,39)(34,42)(36,38)(40,41);(11,14,16,18)(12,19,17,13)(15,22,24,25)(20,26,23,21)(27,30,31,38)(28,34,32,40)(29,36,35,37)(33,41,39,42);(11,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,25)(19,26)(27,28)(29,33)(30,34)(31,32)(35,39)(36,41)(37,42)(38,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612995);