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 | C22×D4 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| prime radical | #3485 | C4 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 418 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)(19,28,30,23)(20,24)(21,26,35,25)(22,27)(29,42,43,36)(31,41,34,37)(32,39,46,38)(33,40)(44,48,45,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,29)(21,34)(22,31)(24,36)(25,28)(26,41)(27,37)(30,35)(32,45)(33,44)(38,42)(39,48)(40,47)(43,46);(3,9)(6,12)(10,15)(13,16)(17,20,22,33)(18,24,27,40)(19,29,31,44)(21,32,35,46)(23,36,37,47)(25,38,26,39)(28,42,41,48)(30,43,34,45)Polycyclic presentation:
2^9|a^2=e,c^2=i,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,e^b=efi,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, 57082);