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 | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
441 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 441 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)(9,10,12,18,14,20,29,35,13,19,28,17,30,37,27,16)(11,21,22,39,25,36,26,15,24,32,23,34,31,38,40,33);(2,6,8,7)(9,11)(10,15,19,33)(12,26)(13,24)(14,31)(16,21,35,32)(17,36,18,38)(20,39,37,34)(22,29)(23,27)(25,30)(28,40);(10,16)(12,27)(14,30)(15,21)(17,20)(18,37)(19,35)(22,40)(23,26)(25,31)(28,29)(32,33)(34,38)(36,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,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, 45932);