Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3493 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3002 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)(11,12,16,20)(13,17,23,30)(14,18,25,32)(15,26,27,19)(21,33,35,28)(22,36,37,29)(24,38,39,31)(34,40,42,41);(3,4)(5,10)(6,8)(7,9)(11,13)(12,17)(14,21)(15,22)(16,23)(18,28)(19,29)(20,30)(24,34)(25,35)(26,36)(27,37)(31,40)(32,33)(38,41)(39,42);(11,14,16,25)(12,18,20,32)(13,21,23,35)(15,24,27,39)(17,28,30,33)(19,31,26,38)(22,34,37,42)(29,40,36,41);(4,8)(7,10)(11,15,16,27)(12,19,20,26)(13,22,23,37)(14,24,25,39)(17,29,30,36)(18,31,32,38)(21,34,35,42)(28,40,33,41)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254131);