Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18996 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3529 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,7)(5,9)(8,10)(11,12,17,22)(13,23,27,33)(14,28,30,36)(15,21,16,20)(18,25,34,26)(19,31,35,29)(24,32,40,42)(37,39,41,38);(5,9)(7,10)(11,13)(12,18)(14,24)(15,25)(16,26)(17,27)(19,32)(20,33)(21,23)(22,34)(28,41)(29,38)(30,40)(31,39)(35,42)(36,37);(3,4)(5,10)(6,8)(7,9)(11,14)(12,19)(13,24)(15,29)(16,31)(17,30)(18,32)(20,28)(21,36)(22,35)(23,37)(25,38)(26,39)(27,40)(33,41)(34,42);(11,15,17,16)(12,20,22,21)(13,25,27,26)(14,29,30,31)(18,33,34,23)(19,28,35,36)(24,38,40,39)(32,41,42,37)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258572);