Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7221 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #340 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C42×C8 | C22 |
| socle | C22 | #1277 |
| prime radical | C23 | #341 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,20,27,23,29,38,46,24,30,39,28,22,40,21,26)(19,31,33,44,36,42,48,47,37,45,32,25,35,41,34,43);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,21,34,22,35,39,32,24,37,38,48,23,36,20,33)(18,25,28,42,29,43,26,45,30,44,27,41,40,47,46,31)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=f,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,c^b=ci,d^b=dg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 508);