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 | |
4800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,5,4,7)(10,14)(15,16)(21,29)(25,30)(26,31)(28,32);(1,3,6,4)(2,7,8,5)(9,10,13,14)(11,15,12,16)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32);(2,8)(3,4)(18,23)(20,27)(25,30)(28,32);(1,2,6,8)(3,7,4,5)(9,11,13,12)(10,15,14,16)(17,19,22,24)(18,20,23,27)(21,26,29,31)(25,28,30,32);(1,4,6,3)(2,5,8,7)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32);(1,5)(2,3)(4,8)(6,7)(9,12,13,11)(10,16,14,15)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618068);