Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 417 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,11,16,12,17,30,34,10,18,28,35,9,15,20,32)(7,19,22,27,21,13,31,8,14,25,29,23,26,33,36,24);(5,7,20,31,12,21,11,29,10,14,30,36,9,26,28,22)(6,13,32,27,17,25,16,8,18,33,34,23,15,19,35,24);(1,2,3,4)(5,8)(6,14)(7,18)(9,27)(10,24)(11,19)(12,23)(13,30)(15,21)(16,22)(17,26)(20,33)(25,28)(29,35)(31,34)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=eg,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dgi,e^b=ei,f^b=fgi,g^b=gi,d^c=di,f^c=fi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47855);