Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150484 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11775 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| prime radical | #28595 | C2 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2930 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
296 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11)(3,9,16,10)(4,18,14,27)(6,12,22,23)(7,13,21,28)(8,17,24,31)(15,26,20,29)(19,25,30,32);(2,8)(3,14)(4,19)(5,21)(7,20)(9,23)(10,13)(11,27)(12,25)(15,22)(16,31)(18,29)(24,28)(26,32);(1,3,7,17,6,14,20,30)(2,9,13,24,12,27,29,32)(4,15,19,5,16,21,31,22)(8,23,18,26,25,11,10,28)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=gi,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cfh,d^b=dg,e^b=efg,g^b=gi,d^c=dhi,f^c=fi,e^d=eh,f^e=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57680);