Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150567 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | C23 | C2×#99 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 56 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)(3,4)(7,8)(9,10,12,18,14,16,25,34)(11,20,22,36,21,32,38,40)(13,17,28,35,29,19,26,37)(15,30,31,23,33,24,39,27);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,15,16,33)(12,24,25,30)(13,27,29,23)(17,32,19,20)(18,36,34,40)(22,26,38,28)(31,37,39,35);(1,3,5,4)(2,6,7,8)(10,16)(11,21)(12,25)(17,19)(23,27)(26,28)(31,39)(36,40)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,e^c=ei,e^d=ehi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57881);