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 | #33675 | 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 | |
400 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,4,7,5,8,3,6)(9,10,15,20,32,24,13,31,16,21,34,23,17,11,14,19)(12,25,28,37,35,39,27,38,29,40,33,18,30,36,26,22);(2,6,8,7)(9,11,17,10,16,24,32,21)(12,22,35,18,29,38,30,37)(13,31,15,23,14,19,34,20)(25,28,36,27,40,33,39,26);(9,12)(10,18)(11,22)(13,28)(14,33)(15,27)(16,29)(17,35)(19,39)(20,25)(21,37)(23,40)(24,38)(26,34)(30,32)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=h,f^2=gh,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, 45929);