Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138850 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 343 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,6,8,3)(9,10,13,18,28,37,29,38,15,20,12,17,14,19,30,39)(11,21,23,33,31,36,26,16,25,32,22,40,24,35,27,34);(1,3,4,6,5,7,8,2)(9,11)(10,16)(12,26)(13,27)(14,31)(15,25)(17,35)(18,36)(19,40)(20,34)(21,38)(22,29)(23,30)(24,28)(32,39)(33,37);(9,11)(10,16)(12,27)(13,26)(14,31)(15,25)(17,36)(18,35)(19,40)(20,34)(21,39)(22,30)(23,29)(24,28)(32,38)(33,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfghi,d^2=ghi,e^2=ghi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=egh,f^c=fghi,h^c=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46074);