Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60536 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(19,25)(22,28)(26,31)(29,32);(2,8)(7,15)(9,13)(10,16)(17,19,20,26,21,25,27,31)(18,22,24,29,23,28,30,32);(18,23)(22,28)(24,30)(29,32);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=di,f^a=fh,c^b=ci,f^b=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251899);