Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150425 |
| 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 | C24 | #99 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,13,19,15,21,12,17)(11,22,24,40,26,35,23,37)(14,20,28,39,32,18,30,38)(16,29,34,31,36,27,33,25);(1,2)(3,7)(4,6)(5,8)(9,11,15,26)(10,16,21,36)(12,27,13,29)(14,31,32,25)(17,37,19,40)(18,35,20,22)(23,28,24,30)(33,38,34,39);(9,12,15,13)(10,17,21,19)(11,23,26,24)(14,30,32,28)(16,33,36,34)(18,39,20,38)(22,37,35,40)(25,27,31,29)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cfh,d^b=dg,e^b=efgi,f^b=fi,d^c=dhi,f^c=fi,e^d=ehi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57632);