Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #388 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,12,15,14,18,13,17)(11,19,20,24,22,23,21,16);(2,4)(5,6)(7,8)(10,15,18,17)(11,20,22,21);(6,8)(9,11,13,21,14,22,12,20)(10,16,15,19,18,24,17,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^c=di,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235489, #72238811, #72238843, … |
| maximal normal subgroup of | #72235489, #72238811, #72238843, … |
| maximal quotient of | #72235540, C2×#123989, #72240026 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31185);