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×#56893 |
| 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 | |
4500 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)(6,9)(8,10)(11,12,16,19)(13,15,18,24)(14,17,23,26)(20,22,21,25)(27,28,32,35)(29,36,33,30)(31,34,40,39)(37,42,41,38);(4,5)(6,10)(13,18)(15,24)(20,21)(22,25)(27,28,32,35)(29,30,33,36)(31,39,40,34)(37,42,41,38);(4,5)(8,9)(11,13,16,18)(12,15,19,24)(14,21,23,20)(17,25,26,22)(27,29)(28,30)(31,37)(32,33)(34,38)(35,36)(39,42)(40,41);(3,5,7,4)(6,9,10,8)(11,14)(12,17)(13,20)(15,22)(16,23)(18,21)(19,26)(24,25)(27,30)(28,33)(29,35)(31,38)(32,36)(34,41)(37,39)(40,42);(3,6,7,10)(4,8,5,9)(11,15)(12,18)(13,19)(14,22)(16,24)(17,21)(20,26)(23,25)(27,31,32,40)(28,34,35,39)(29,37,33,41)(30,38,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007203);