Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149899 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #23909 | C2 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
630 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 630 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,17,24,33)(7,18,20,27,21,32,35,29)(8,13,22,31,10,16,26,14)(12,19,28,34,30,25,36,23);(5,7)(6,12,13,27,17,30,16,29)(8,19)(9,23)(10,25)(11,21)(14,28,33,18,31,36,15,32)(20,22)(24,34)(26,35);(2,4)(5,8,11,10)(6,13,17,16)(7,19,21,25)(9,22,24,26)(12,27,30,29)(14,15,31,33)(18,28,32,36)(20,34,35,23)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,f^b=fg,g^b=gh,f^d=fhi,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59851);