Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C4×#3067 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#121 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4);(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,19)(13,23)(18,32)(20,28)(21,27)(24,29)(25,30)(26,34)(31,33)(35,36);(5,7,9,19,11,21,10,20)(6,12,15,17,16,28,14,27)(8,18,23,33,24,34,25,35)(13,26,22,31,30,32,29,36);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,35)(21,34)(27,36)(28,32);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,24,25)(12,27,28,17)(13,29,30,22)(18,33,34,35)(26,36,32,31)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280105);