Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33767 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1622 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)(6,8)(7,10)(13,21)(14,24)(15,26)(17,29)(19,34)(20,33)(22,39)(23,38)(25,28)(27,37)(30,41)(31,42)(32,40)(35,36);(3,5)(4,7)(6,10)(8,9)(11,12)(13,17)(14,19)(15,20)(16,18)(21,29)(22,31)(23,30)(24,34)(25,35)(26,33)(27,32)(28,36)(37,40)(38,41)(39,42);(5,10)(7,9)(11,13,16,23)(12,17,18,30)(14,25,27,39)(15,28,26,22)(19,35,32,42)(20,36,33,31)(21,37,38,24)(29,40,41,34);(4,8)(7,9)(12,18)(17,30)(19,32)(20,33)(29,41)(31,36)(34,40)(35,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530496);