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 | C23 | #447 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 403 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10,14,20,29,38,13,17,16,22,12,19,15,21,30,34)(11,23,26,18,32,33,31,40,28,39,24,37,27,35,25,36);(2,6,8,7)(10,17)(12,13)(14,30)(15,29)(18,33)(19,21)(20,38)(22,34)(23,40)(24,31)(25,26)(27,32)(35,37)(36,39);(9,11,16,28)(10,18,22,37)(12,25,14,31)(13,26,30,24)(15,32,29,27)(17,33,34,35)(19,23,20,39)(21,40,38,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=i,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, 45933);