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×#54287 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4702 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);(4,7)(5,9)(11,12,16,19)(13,14,21,17)(15,18,23,25)(20,22,26,24)(27,28,32,35)(29,30,37,33)(31,39,40,34)(36,41,42,38);(4,7)(8,10)(15,23)(18,25)(20,26)(22,24)(27,29,32,37)(28,33,35,30)(31,36,40,42)(34,41,39,38);(3,4,6,7)(5,8,9,10)(11,13,16,21)(12,17,19,14)(15,20,23,26)(18,24,25,22)(27,29)(28,33)(30,35)(31,36)(32,37)(34,41)(38,39)(40,42);(3,5,6,9)(4,8,7,10)(11,14,16,17)(12,13,19,21)(15,22,23,24)(18,20,25,26)(27,30,32,33)(28,29,35,37)(31,38,40,41)(34,36,39,42);(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,31,32,40)(28,34,35,39)(29,36,37,42)(30,38,33,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010109);