Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#2098 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#399 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,9,14,10,15,11,25)(7,16,18,28,20,32,19,12)(8,21,23,31,24,13,26,30)(17,33,22,36,34,29,35,27);(5,7,11,18,10,20,9,19)(6,12,25,16,15,28,14,32)(8,22,26,34,24,35,23,17)(13,29,31,27,21,33,30,36);(5,8)(6,13)(7,17)(9,23)(10,24)(11,26)(12,27)(14,30)(15,21)(16,33)(18,22)(19,35)(20,34)(25,31)(28,36)(29,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266511);