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×#57702 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4620 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);(4,8)(6,9)(11,12,15,16)(13,19,21,17)(14,22,23,18)(20,25,24,26)(27,28)(29,34)(30,36)(31,33)(32,35)(37,41)(38,42)(39,40);(3,4)(5,9)(6,10)(7,8)(12,16)(14,23)(17,19)(20,24)(27,29,33,30)(28,34,31,36)(32,38,40,37)(35,42,39,41);(4,8)(5,10)(11,13)(12,17)(14,24)(15,21)(16,19)(18,26)(20,23)(22,25)(27,30,33,29)(28,34,31,36)(32,38,40,37)(35,41,39,42);(3,5)(4,6)(7,10)(8,9)(11,14,15,23)(12,18,16,22)(13,20,21,24)(17,25,19,26)(27,31,33,28)(29,36,30,34)(32,39,40,35)(37,42,38,41);(3,6)(4,5)(7,9)(8,10)(11,12)(13,17)(14,18)(15,16)(19,21)(20,25)(22,23)(24,26)(27,32)(28,35)(29,37)(30,38)(31,39)(33,40)(34,41)(36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008667);