Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150429 |
| 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 | |
296 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 62 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)(9,10,13,19,15,17,12,28)(11,21,25,16,27,33,40,35)(14,20,30,36,22,37,31,18)(23,39,24,34,26,38,29,32);(1,2,5,8)(3,6,4,7)(9,11,22,23,15,27,14,26)(10,16,20,34,17,35,37,32)(12,29,31,40,13,24,30,25)(18,33,28,39,36,21,19,38);(1,3,5,4)(2,7,8,6)(10,17)(11,23)(14,22)(16,32)(19,28)(21,38)(24,25)(26,27)(29,40)(30,31)(33,39)(34,35)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=gi,d^2=ghi,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfhi,d^b=dg,e^b=efgh,d^c=dhi,f^c=fi,e^d=ehi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58119);