Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153213 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2226 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
215 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,14,20,7)(8,24)(9,27)(10,26)(11,32)(12,33)(15,36)(16,28,22,21)(17,31,30,18)(19,23,35,34)(25,29);(5,6,15,17,11,18,8,14,33,35,24,16,25,28,36,23,13,20,10,30,29,31,27,7,12,19,9,22,32,21,26,34);(2,4)(5,7)(6,8)(9,28)(10,31)(11,30)(12,34)(13,14)(15,18)(16,25)(17,29)(19,36)(20,27)(21,24)(22,32)(23,33)(26,35)Polycyclic presentation:
2^9|a^2=de,b^2=dgh,d^2=ghi,e^2=gi,f^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fghi,g^a=gh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,d^c=dg,e^c=eghi,f^c=fgh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60549);