Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150198 |
| 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 | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | #537 | C2×C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
370 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 370 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:
(2,6,7,8)(9,10,12,17,14,19,27,37)(11,20,23,31,21,33,39,15)(13,18,29,16,30,34,26,35)(22,38,28,36,25,32,24,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,25)(13,28)(14,21)(16,33)(17,36)(18,38)(19,31)(20,35)(22,27)(23,26)(24,30)(29,39)(32,34)(37,40);(1,3,5,4)(2,6,7,8)(11,21)(15,31)(20,33)(22,25)(23,39)(24,28)(32,38)(36,40)Polycyclic presentation:
2^9|a^2=e,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,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, 57521);