Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5,7,8)(4,6)(9,10,14,20,15,21,34,25,16,22,35,39,13,19,24,37)(11,23,18,38,26,12,28,30,27,29,32,40,31,17,36,33);(1,3)(2,5,6,8)(4,7)(9,11,24,28,13,31,35,18,16,27,34,36,15,26,14,32)(10,17,37,38,19,29,39,33,22,12,25,40,21,23,20,30);(9,12)(10,18)(11,25)(13,23)(14,33)(15,29)(16,17)(19,36)(20,31)(21,28)(22,32)(24,40)(26,39)(27,37)(30,35)(34,38)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dgi,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44869);