Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187878 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62834 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,5,6,4,3,7,8)(9,10,12,14,13,11,15,16);(5,7)(6,8)(9,11)(10,13)(12,14)(15,16);(1,3)(2,4)(5,6)(7,8)(12,15)(14,16);(2,3)(6,8)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226146, #72226253, #72226270, … |
| maximal normal subgroup of | #72226146, #72226253, #72226270, … |
| maximal quotient of | #72338860 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95114);