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 | C2×#532 | 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 | |
642 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 642 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)(5,6,11,16,12,15,29,31,10,17,28,35,9,26,19,34)(7,18,21,27,22,13,30,8,14,25,33,23,20,32,36,24);(5,7,19,36,12,22,11,21,10,14,29,30,9,20,28,33)(6,13,31,27,15,25,35,8,17,32,34,23,26,18,16,24);(2,4)(5,8)(6,14)(7,17)(9,27)(10,24)(11,18)(12,23)(13,29)(15,20)(16,21)(19,32)(22,26)(25,28)(30,31)(33,35)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=df,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, 44901);