Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C8 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | #1889 |
| prime radical | C2×2−1+4 | C2×C4 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
525 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,4,7,5,3,6,8)(9,10,14,19)(11,20)(12,17,21,30)(13,18)(15,22)(16,28)(23,33)(24,31,27,40)(25,36)(26,35)(29,34)(32,39,37,38);(1,3,4,8,5,2,6,7)(9,11,21,23,14,25,12,22)(10,15,30,20,19,33,17,36)(13,27,28,38,29,24,26,39)(16,31,34,37,35,40,18,32);(9,12,14,21)(10,16,19,35)(11,22,25,23)(13,28,29,26)(15,31,33,40)(17,18,30,34)(20,37,36,32)(24,39,27,38)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,e^a=egi,g^a=gi,d^b=di,e^b=egi,f^b=fi,d^c=dg,e^c=ei,f^c=fg,e^d=ei,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44022);