Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,5)(8,15)(9,10)(14,16)(17,18,22,26,23,28,21,27)(19,29,25,24,31,20,30,32);(1,2,5,9,6,10,4,7)(3,8,13,15,11,16,12,14)(17,19,21,30,23,31,22,25)(18,24,27,29,28,32,26,20);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,15)(10,16)(17,20,22,32,23,29,21,24)(18,25,26,31,28,30,27,19);(3,11)(8,16)(12,13)(14,15)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=ch,e^a=eh,g^a=ghi,e^b=ei,d^c=dhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90219);