Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150788 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C24 | D16 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
271 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2)(3,7,8,5)(4,6)(10,17,19,37)(11,23)(12,28)(13,25)(14,30)(15,34)(18,22,39,36)(20,40,21,24)(26,27)(29,33,38,35)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10,16,19)(11,24,27,40)(12,29,32,38)(13,33,14,35)(15,36,34,22)(17,28,37,31)(18,26,39,23)(20,30,21,25);(3,8)(5,7)(9,11,25,28,15,12,14,23,16,27,30,31,34,32,13,26)(10,18,33,29,22,37,21,24,19,39,35,38,36,17,20,40)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=ghi,d^2=gh,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=cfi,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58067);