Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#28179 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12)(13,18)(14,26)(15,31)(16,20)(17,21)(19,28)(22,35)(23,33)(24,40)(25,34)(27,41)(29,36)(30,37)(32,39)(38,42);(4,7)(8,9)(11,13,16,25,17,23,15,24)(12,18,20,34,21,33,31,40)(14,27,29,38,30,39,28,22)(19,35,26,41,36,42,37,32);(11,14,17,30)(12,19,21,36)(13,22,23,38)(15,29,16,28)(18,32,33,41)(20,37,31,26)(24,27,25,39)(34,42,40,35);(3,5,6,10)(4,8,7,9)(13,23)(18,33)(22,38)(24,25)(27,39)(32,41)(34,40)(35,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370088);