Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33651 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1604 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)(5,8)(6,7)(9,10)(11,12,16,21)(13,17,22,30)(14,18,27,34)(15,28,26,20)(19,35,33,25)(23,29,36,40)(24,39,37,32)(31,42,41,38);(4,7)(8,10)(11,13)(12,17)(14,23)(15,24)(16,22)(18,29)(19,31)(20,32)(21,30)(25,38)(26,37)(27,36)(28,39)(33,41)(34,40)(35,42);(3,5,6,9)(4,8,7,10)(12,18)(14,25)(15,26)(17,29)(19,20)(21,34)(23,38)(24,37)(27,35)(28,33)(30,40)(31,32)(36,42)(39,41);(13,22)(17,30)(23,36)(24,37)(29,40)(31,41)(32,39)(38,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530238);