Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #354 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #384 | C23 |
| socle | C22 | C2×#354 |
| prime radical | C23 | C2×#99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
285 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,20,23,30)(12,16)(13,18,27,17)(14,19)(15,22,29,26)(21,34,36,40)(24,32,38,31)(25,33)(28,35,39,37);(9,11,14,22)(10,15)(12,21,25,35)(13,26,27,23)(16,28)(17,20)(18,29)(19,30)(24,37,38,36)(31,34)(32,39)(33,40);(1,3,5,4)(2,6,8,7)(9,12,14,25)(10,16,19,33)(11,21,22,35)(13,24,27,38)(15,28,30,40)(17,31,18,32)(20,34,29,39)(23,36,26,37)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=gi,d^2=hi,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=dh,e^b=efh,f^b=fi,e^d=ehi,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56898);