Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163298 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
837 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)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23)(25,26,31,33,29,28,30,38)(27,34,37,40,35,32,36,39);(1,3,4,8)(2,5,6,7)(25,27,29,35)(26,32,28,34)(30,36,31,37)(33,39,38,40);(3,8)(5,7)(9,11)(10,14)(12,20)(13,19)(15,22)(16,24)(17,23)(18,21)(25,28,29,26)(27,32,35,34)(30,33,31,38)(36,40,37,39);(2,6)(5,7)(10,15)(14,22)(16,23)(17,24)(27,35)(32,34)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,d^b=dh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 122756);