Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13681 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C2×#555 | C22 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
507 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 507 conjugacy classes of subgroups — too many to draw. Only its 62 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)(5,6,11,16,13,7,9,15)(8,14,22,30,24,18,20,29)(10,19,26,35,12,17,27,33)(21,36,28,34,23,31,25,32);(1,2,3,4)(5,7,13,6)(8,18,24,14)(9,15,11,16)(10,17,12,19)(20,29,22,30)(21,31,23,36)(25,32,28,34)(26,35,27,33);(5,8,13,24)(6,14,7,18)(9,25,11,28)(10,23,12,21)(15,32,16,34)(17,36,19,31)(20,26,22,27)(29,35,30,33)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eghi,g^a=gh,h^a=hi,d^b=di,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,e^c=eh,f^c=fg,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59785);