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 | #391 | C23 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
441 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 441 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,21,24,31)(12,17)(13,28)(14,19,30,36)(15,20)(16,23,32,40)(18,29,37,27)(22,33,39,34)(25,35,26,38);(9,11,14,25,15,26,30,24)(10,16,19,33,20,34,36,32)(12,23,27,39,13,22,29,40)(17,35,18,21,28,31,37,38);(2,5)(6,8)(9,12,15,13)(10,17,20,28)(11,22,26,23)(14,27,30,29)(16,31,34,35)(18,36,37,19)(21,33,38,32)(24,39,25,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=i,e^2=g,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dhi,e^b=ef,h^b=hi,e^d=ehi,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57032);