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×#56077 |
| 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 | |
4598 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)(5,9)(6,7)(8,10)(11,12)(13,16)(14,19)(15,20)(17,21)(18,23)(22,24)(25,26)(27,28,31,34)(29,35,36,32)(30,33,38,40)(37,41,42,39);(3,5,7,10)(4,8,6,9)(12,13)(14,21)(18,20)(22,26)(29,36)(32,35)(37,42)(39,41);(5,10)(8,9)(11,13,16,12)(14,19,21,17)(15,20,23,18)(22,25,26,24)(27,29,31,36)(28,32,34,35)(30,37,38,42)(33,39,40,41);(3,5)(4,9)(6,8)(7,10)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40)(41,42);(3,6,7,4)(5,8,10,9)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,30,31,38)(28,33,34,40)(29,37,36,42)(32,39,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009928);