Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C4 | C22×C32 |
| Frattini, cosocle | C4×C16 | C23 |
| socle, prime radical | C22 | D4×C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,20,25,21,26,31,38,22,27,32,39,33,40,44,47,23,19,24,28,34,29,36,41,35,30,37,42,45,43,46,48);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9);(2,7)(6,15)(8,12)(9,16)(17,19,20,28,21,29,31,41,22,30,32,42,33,43,44,48,23,18,24,25,34,26,36,38,35,27,37,39,45,40,46,47)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=cf,d^a=df,d^b=df176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58294);