Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#102 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,21,28,24,30,23,26)(19,31,34,41,32,46,36,48)(20,29,37,45,38,44,22,27)(25,39,40,47,42,35,43,33);(1,3)(2,6,8,15)(4,11)(5,10)(7,9,12,16)(13,14)(17,19,22,35,23,36,38,47,24,32,37,39,21,34,20,33)(18,25,29,41,28,40,45,46,30,42,44,48,26,43,27,31);(3,10)(6,15)(9,16)(11,14)(18,26,30,28)(19,32)(25,40,42,43)(27,44,45,29)(31,41,46,48)(33,47)(34,36)(35,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi,h^a=hi,c^b=cgh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35812);