Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×#1395 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #99 |
| commutator | C2×C4 | C43 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | C2×#342 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4);(5,6)(7,15,17,27)(8,19,20,12)(9,22)(10,14)(11,16)(13,23)(18,32,31,28)(21,26,34,33)(24,36)(25,30)(29,35);(5,7,9,18,10,17,23,31)(6,11,13,24,14,25,22,35)(8,16,21,29,20,30,34,36)(12,15,26,28,19,27,33,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53558);