Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
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 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
543 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 543 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2)(3,4)(5,6,11,16,12,15,13,27,10,7,17,21,9,20,18,28)(8,14,24,31,22,32,34,36,23,19,26,33,25,30,29,35);(5,7,18,27,12,20,11,21,10,6,13,28,9,15,17,16)(8,19,34,35,22,30,26,31,23,14,29,36,25,32,24,33);(2,4)(5,8)(6,14)(7,19)(9,25)(10,23)(11,26)(12,22)(13,29)(15,32)(16,33)(17,24)(18,34)(20,30)(21,31)(27,35)(28,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,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=dg,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, 44946);