Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131820 |
| 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 | #61406 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
631 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 631 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,6,7,4,3,5,8)(9,10,14,21,16,20,12,28,15,22,31,40,13,32,30,19)(11,23,26,17,25,18,24,37,27,39,29,36,33,38,34,35);(5,6)(7,8)(9,11)(10,17)(12,29)(13,25)(14,34)(15,27)(16,33)(18,19)(20,35)(21,23)(22,36)(24,31)(26,30)(28,38)(32,37)(39,40);(1,3,4,2)(5,7,6,8)(9,11)(10,18)(12,24)(13,33)(14,26)(15,27)(16,25)(17,19)(20,39)(21,37)(22,38)(23,32)(28,36)(29,31)(30,34)(35,40)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=hi,e^2=g,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=eg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39090);