Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C2×C44 |
| direct product | C4×#1747 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C42 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#376 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1276 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)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,23,31)(16,27,30,35)(19,24)(20,25)(21,26)(22,29,33,36)(32,34);(5,7)(6,11,14,23)(8,16)(9,17)(10,18)(12,22,25,33)(13,15,26,28)(19,29)(20,30)(21,31)(24,27,34,35)(32,36);(5,8,9,19,10,20,21,32)(6,12,13,24,14,25,26,34)(7,16,17,29,18,30,31,36)(11,22,15,27,23,33,28,35)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260368);