Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×#2153 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×C4⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
852 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);(5,6,9,14,10,13,23,28,11,15,24,29,8,21,22,34)(7,16,19,31,20,26,33,35,17,30,32,36,18,12,25,27);(5,7,8,18,11,17,10,20)(6,12,13,16,15,26,21,30)(9,19,22,25,24,32,23,33)(14,27,28,31,29,35,34,36);(7,17)(12,26)(16,30)(18,20)(19,32)(25,33)(27,35)(31,36)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400295);