Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(11,17)(12,20)(14,22)(16,23)(18,24)(19,21)(25,26,31,33,29,28,30,38)(27,34,37,40,35,32,36,39);(1,3)(2,5)(4,8)(6,7)(9,10,13,15)(11,18,19,14)(12,16,20,23)(17,24,21,22)(25,27,30,36,29,35,31,37)(26,32,38,40,28,34,33,39);(3,8)(5,7)(9,11,13,19)(10,14,15,18)(12,21,20,17)(16,24,23,22)(25,28,29,26)(27,34,35,32)(30,33,31,38)(36,39,37,40);(2,6)(5,7)(10,15)(14,18)(16,23)(22,24)(26,28)(32,34)(33,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,d^b=dhi,d^c=di,f^c=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 125006);