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×#55980 |
| 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 | |
4750 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)(5,10)(6,9)(7,8)(11,12,15,18)(13,19,20,16)(14,17,22,24)(21,25,26,23)(28,32)(29,35)(30,38)(40,42);(3,5)(4,9)(6,8)(7,10)(13,20)(16,19)(21,26)(23,25)(27,28,31,32)(29,36,35,33)(30,34,38,39)(37,42,41,40);(5,6)(9,10)(11,13)(12,16)(14,21)(15,20)(17,23)(18,19)(22,26)(24,25)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39)(41,42);(3,6,8,5)(4,9,7,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,29,31,35)(28,33,32,36)(30,37,38,41)(34,40,39,42);(3,7,8,4)(5,9,6,10)(11,14,15,22)(12,17,18,24)(13,21,20,26)(16,23,19,25)(27,30)(28,34)(29,37)(31,38)(32,39)(33,40)(35,41)(36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010045);