Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149901 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×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 | #3497 | C4 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 423 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,8,5,6,3,7)(9,10)(11,19,22,37,20,35,23,38)(12,24)(13,28)(14,16)(15,21,31,39,33,25,32,40)(17,26)(18,30)(27,36)(29,34);(2,6)(3,4)(9,11)(10,15)(12,25)(13,22)(14,20)(16,33)(17,35)(18,31)(19,34)(21,27)(23,30)(24,38)(26,40)(28,32)(29,39)(36,37);(1,3,5,4)(2,7,6,8)(10,16)(11,20)(17,34)(18,28)(21,25)(22,23)(24,36)(39,40)Polycyclic presentation:
2^9|a^2=e,c^2=ghi,d^2=i,e^2=g,f^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=cgh,d^b=di,e^b=ef,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57163);