Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149889 |
| 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 | C22×Q8 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | #3487 | 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 | |
415 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,29,32,39)(20,25)(21,36)(22,27,38,44)(23,28)(24,31,40,48)(26,37,46,35)(30,41,47,42)(33,43,34,45);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,31)(21,30)(22,32)(23,34)(25,43)(26,45)(27,40)(28,42)(29,46)(33,38)(35,48)(36,39)(37,47)(41,44);(3,9)(6,12)(10,15)(13,16)(17,20,23,21)(18,25,28,36)(19,30,34,31)(22,35,38,37)(24,39,42,43)(26,44,46,27)(29,41,45,40)(32,47,33,48)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=i,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgi,d^b=di,e^b=ef,f^b=fi,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, 57088);