Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136514 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #450 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
291 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(2,6)(3,7)(4,8)(9,10,14,18,15,20,12,16)(11,21,22,38,24,39,25,36)(13,27,29,37,30,32,26,19)(17,33,31,23,35,28,34,40);(1,2,5,3)(4,7,8,6)(10,16,20,18)(17,31,35,34)(19,32,37,27)(21,38,39,36);(1,3,5,2)(4,6,8,7)(9,11,15,24)(10,17,20,35)(12,22,14,25)(13,28,30,33)(16,31,18,34)(19,36,37,38)(21,27,39,32)(23,26,40,29)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=g,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dgi,e^b=egi,d^c=dgi,e^c=ehi,f^c=fi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43745);