Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2128 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #407 | C23 |
| socle | C2 | #11786 |
| prime radical | #525 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2929 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(2,8,10,24)(3,12)(4,17,20,22)(9,26,27,23)(11,18,21,25)(13,29,28,30)(14,31,19,32)(15,16);(1,2,5,21,7,10,6,11)(3,13,15,9,12,28,16,27)(4,18,17,24,20,25,22,8)(14,26,32,29,19,23,31,30);(1,3,6,16,7,12,5,15)(2,9,21,28,10,27,11,13)(4,19,17,31,20,14,22,32)(8,23,25,29,24,26,18,30)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=gh,d^2=ghi,f^2=i,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efghi,f^b=fi,h^b=hi,d^c=dhi,e^d=ehi,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57727);