Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343406 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62800 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,24)(20,22)(21,23)(25,28)(26,30)(27,29)(31,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,22)(20,28)(23,25)(24,29)(27,32)(30,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,23)(22,25)(24,30)(29,31);(17,19)(18,24)(20,27)(21,26)(22,29)(23,30)(25,31)(28,32);(19,26)(24,30)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=g,c^2=hi,f^2=i,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250609);