Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#2166 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
670 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);(5,6,8,14,9,15,22,31,10,13,21,29,23,12,26,28)(7,16,18,30,19,27,33,36,17,11,20,25,32,24,34,35);(5,7)(6,11)(8,20)(9,19)(10,17)(12,27)(13,16)(14,30)(15,24)(18,21)(22,34)(23,32)(25,29)(26,33)(28,35)(31,36);(6,12,13,15)(7,17)(8,21)(11,24,16,27)(14,31,29,28)(19,32)(22,26)(25,36,30,35)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400376);