Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#25596 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3424 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);(3,4)(5,9)(6,7)(8,10)(11,12,15,19)(13,21,23,17)(14,18,26,34)(16,29,28,20)(22,31,38,37)(24,32,30,36)(25,39,40,33)(27,42,41,35);(4,7)(8,9)(11,13)(12,17)(14,25)(15,23)(16,30)(18,33)(19,21)(20,36)(22,27)(24,28)(26,40)(29,32)(31,35)(34,39)(37,42)(38,41);(11,14)(12,18)(13,22,23,38)(15,26)(16,27)(17,31,21,37)(19,34)(20,35)(24,25,30,40)(28,41)(29,42)(32,33,36,39);(3,5,6,10)(4,8,7,9)(12,19)(14,26)(17,21)(20,29)(22,38)(25,40)(27,41)(32,36)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356285);