Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149749 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | C2×2−1+4 | C2×C4 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
454 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 70 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)(5,6,9,14,8,11,21,29)(7,16,19,27,17,32,35,12)(10,15,26,13,22,30,24,28)(18,34,25,33,23,31,20,36);(1,2,3,4)(6,11)(12,27)(13,28)(14,29)(15,30)(16,32)(31,34)(33,36);(5,7)(6,12)(8,17)(9,23)(10,25)(11,27)(13,32)(14,33)(15,34)(16,28)(18,21)(19,24)(20,22)(26,35)(29,36)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=di,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^c=ei,f^c=fg,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59410);