Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#2214 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#459 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
690 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);(5,6,12,18)(7,16,21,27)(8,22,24,35)(9,17,11,15)(10,13,20,28)(14,29,31,19)(23,36,26,32)(25,33,34,30);(5,7,11,20,12,21,9,10)(6,13,17,27,18,28,15,16)(8,19,26,34,24,29,23,25)(14,22,33,36,31,35,30,32);(5,8)(6,14)(7,19)(9,23)(10,25)(11,26)(12,24)(13,22)(15,30)(16,32)(17,33)(18,31)(20,34)(21,29)(27,36)(28,35)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403027);