Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#1814 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1520 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)(7,16)(8,21,23,28)(9,24)(10,14)(11,15)(12,18)(13,22,29,25)(17,26,33,36)(19,27)(20,31)(30,32,35,34);(5,7,9,18,11,20,10,19)(6,12,14,16,15,27,24,31)(8,17,22,32,23,33,25,34)(13,26,28,30,29,36,21,35);(5,8)(6,13)(7,17)(9,22)(10,25)(11,23)(12,26)(14,28)(15,29)(16,30)(18,32)(19,34)(20,33)(21,24)(27,36)(31,35)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262153);