Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150584 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| 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 | |
343 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17,14,19,29,35)(11,20,23,37,22,36,39,33)(12,16,25,32,26,18,28,34)(15,27,30,21,31,38,40,24);(2,5)(3,8)(6,7)(9,11)(10,15,19,31)(12,21)(13,27)(14,22)(16,20,18,36)(17,33,35,37)(23,25)(24,26)(28,39)(29,38)(30,34,40,32);(2,6)(5,7)(9,12,14,26)(10,16,19,18)(11,21,22,24)(13,28,29,25)(15,20,31,36)(17,34,35,32)(23,27,39,38)(30,33,40,37)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=i,d^2=gi,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=dg,e^b=efg,f^b=fi,e^c=ei,e^d=ehi,f^e=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57785);