Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×#2178 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
736 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);(5,6,10,14,11,16,26,15)(7,17,19,28,21,34,20,36)(8,13,22,31,23,32,9,24)(12,27,29,35,30,25,33,18);(5,7)(6,12)(8,18)(9,25)(10,19)(11,21)(13,28)(14,29)(15,33)(16,30)(17,24)(20,26)(22,27)(23,35)(31,34)(32,36);(5,8,10,22,11,23,26,9)(6,13,14,31,16,32,15,24)(7,18,19,27,21,35,20,25)(12,28,29,34,30,36,33,17)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400437);