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×#54549 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4436 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);(4,6)(5,7)(11,12,16,20)(13,17,21,26)(14,18,24,23)(15,19,25,22)(27,28)(29,31)(30,37)(32,34)(33,35)(36,39)(38,41)(40,42);(3,4)(5,9)(6,8)(7,10)(13,21)(15,25)(17,26)(19,22)(27,29,33,30)(28,31,35,37)(32,38,40,36)(34,41,42,39);(3,5,8,7)(4,9,6,10)(11,13)(12,17)(14,19)(15,23)(16,21)(18,25)(20,26)(22,24)(27,30,33,29)(28,31,35,37)(32,38,40,36)(34,39,42,41);(3,6,8,4)(5,9,7,10)(11,14)(12,18)(13,22)(15,17)(16,24)(19,21)(20,23)(25,26)(27,31)(28,30)(29,35)(32,39)(33,37)(34,38)(36,42)(40,41);(3,7,8,5)(4,9,6,10)(11,15)(12,19)(13,23)(14,17)(16,25)(18,21)(20,22)(24,26)(27,32,33,40)(28,34,35,42)(29,36,30,38)(31,39,37,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010961);