Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136516 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(2,5)(3,7)(6,8)(9,10,14,20,16,22,15,21)(11,23,26,38,28,39,27,33)(12,29,30,19,31,17,13,32)(18,34,35,25,37,24,36,40);(1,2)(3,8)(4,6)(5,7)(10,17,22,29)(12,30)(13,31)(14,15)(18,33,37,38)(19,21,32,20)(23,35,39,36)(24,40)(25,34)(26,27);(3,7)(5,8)(9,11,16,28)(10,18,22,37)(12,25,31,40)(13,24,30,34)(14,27,15,26)(17,33,29,38)(19,23,32,39)(20,36,21,35)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=i,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dgi,e^c=ehi,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309780, #72309788, #72309793, … |
| maximal normal subgroup of | #72309780, #72309788, #72309793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43740);