Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149885 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator, prime radical | C22×D4 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,9,6,10,4,8)(3,7,11,14,12,15,13,16)(17,18)(19,28,30,23)(20,24)(21,26,34,25)(22,27)(29,41,42,35)(31,40,44,36)(32,38,45,37)(33,39)(43,47,48,46);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,14)(10,15)(17,19,21,30,22,31,34,44)(18,23,26,28,27,36,25,40)(20,29,32,42,33,43,45,48)(24,35,38,41,39,46,37,47);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,15,16)(17,20,22,33)(18,24,27,39)(19,29,31,43)(21,32,34,45)(23,35,36,46)(25,37,26,38)(28,41,40,47)(30,42,44,48)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gi,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=dh,e^b=efi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57221);