Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,13)(14,16)(21,29)(25,30)(26,32)(28,31);(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,25)(22,23)(24,27)(26,28)(29,30)(31,32);(1,2,4,6)(3,5,8,7)(18,23)(20,24)(25,30)(28,31);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32);(1,3)(2,5)(4,8)(6,7)(17,20,22,24)(18,19,23,27)(21,28,29,31)(25,26,30,32);(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)Polycyclic presentation:
2^9|c^2=h,d^2=g,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621723);