Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×C22⋊C8 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,7)(3,8,12,15,13,16,11,14);(2,7,10,9)(8,14,16,15)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(1,3,5,12,6,13,4,11)(2,8,9,15,10,16,7,14)(18,22)(23,28)(24,29)(30,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fg,c^b=ci152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35817);