Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #121 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1797 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9,7,4,5,15)(3,10,13,11,14,16,12,8)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,3,6,13,7,14,5,12)(2,8,15,16,4,11,9,10);(1,4)(2,7)(3,11)(5,15)(6,9)(8,14)(10,13)(12,16)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^c=eg,f^c=fg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41919);