Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3488 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4340 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)(3,4)(9,10,12,15,13,11,16,14);(5,6)(7,8)(9,11)(10,13)(12,14)(15,16);(2,4)(6,8)(9,10)(11,13)(12,14)(15,16);(9,10)(11,13)(12,15)(14,16)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^c=fi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257992, #72316872, #72317709, … |
| maximal normal subgroup of | #72257992, #72316872, #72317709, … |
| maximal quotient of | #72258047, #72291946, #72317089 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266852);