Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136514 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #466 | Q8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,6,9,7,4,5,10)(3,11,13,8,15,16,14,12)(17,18,22,26,23,28,20,24)(19,29,30,46,32,47,33,44)(21,35,37,45,38,40,34,27)(25,41,39,31,43,36,42,48);(1,3,5,14,7,15,6,13)(2,8,10,11,4,12,9,16)(18,24,28,26)(25,39,43,42)(27,40,45,35)(29,46,47,44);(1,4)(2,7)(3,12)(5,9)(6,10)(8,15)(11,13)(14,16)(17,19)(18,25)(20,30)(21,36)(22,33)(23,32)(24,39)(26,42)(27,44)(28,43)(29,40)(31,37)(34,48)(35,47)(38,41)(45,46)Polycyclic presentation:
2^9|a^2=f,b^2=fh,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dgi,e^b=egi,d^c=dgi,e^c=ehi,f^c=fi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43767);