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×#57860 |
| 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 | |
4734 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)(5,7)(6,8)(9,10)(11,12)(13,19)(14,16)(15,22)(17,21)(18,23)(20,24)(25,26)(27,28,32,36)(29,30,34,33)(31,39,41,35)(37,42,40,38);(6,10)(8,9)(12,14)(15,23)(17,19)(20,26)(27,29,32,34)(28,33,36,30)(31,40,41,37)(35,38,39,42);(4,5)(8,9)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,29)(28,30)(31,37)(32,34)(33,36)(35,38)(39,42)(40,41);(3,5,7,4)(6,9,10,8)(11,14,16,12)(13,17,21,19)(15,22,23,18)(20,24,26,25)(27,30)(28,34)(29,36)(31,38)(32,33)(35,40)(37,39)(41,42);(3,6,7,10)(4,8,5,9)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,31,32,41)(28,35,36,39)(29,37,34,40)(30,38,33,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008753);