Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131804 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 599 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,6,7,4,3,5,8)(9,10,13,18,14,23,29,37,16,19,30,28,15,11,20,22)(12,24,25,38,31,34,40,39,27,21,35,36,26,17,32,33);(5,6)(7,8)(9,11)(10,15)(12,21)(13,28)(14,19)(16,23)(17,26)(18,30)(20,22)(24,27)(25,39)(29,37)(31,34)(32,36)(33,35)(38,40);(1,3,4,2)(5,7,6,8)(9,12,16,27)(10,17,19,34)(11,21,23,24)(13,25,30,35)(14,31,15,26)(18,33,28,39)(20,32,29,40)(22,36,37,38)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=hi,e^2=gi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,e^c=egi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39087);