Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #120 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1436 |
| prime radical | C24 | C4×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10)(11,20)(12,25)(13,29)(14,18)(15,19)(16,22)(17,27)(21,36)(23,32)(24,31)(26,34)(28,33)(30,38)(35,37)(39,40);(2,6)(3,7)(9,11,14,22,15,24,13,23)(10,16,18,31,19,32,29,20)(12,21,27,37,28,38,26,39)(17,30,33,40,34,36,25,35);(1,3,5,6,4,7,8,2)(9,12)(10,17)(11,21)(13,26)(14,27)(15,28)(16,30)(18,33)(19,34)(20,35)(22,37)(23,39)(24,38)(25,29)(31,40)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ci,d^b=dh,e^c=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16725);