Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138616 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33669 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
452 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10,13,18,15,20,28,36,14,19,12,26,29,38,30,17)(11,21,23,35,25,37,31,34,24,33,22,39,32,40,27,16);(2,6,8,7)(9,11)(10,16,19,34)(12,27)(13,31)(14,24)(15,32)(17,33,36,21)(18,37,26,40)(20,39,38,35)(22,30)(23,28)(25,29);(10,17)(12,28)(13,30)(15,29)(16,33)(18,38)(19,36)(20,26)(21,34)(22,31)(23,27)(25,32)(35,37)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=h,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45930);