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 | |
4759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(4,8)(9,10,13,12)(11,15,16,14)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28);(2,6)(3,7)(9,11,13,16)(10,14,12,15)(18,23)(20,27)(25,29)(28,32);(1,2)(3,4)(5,6)(7,8)(9,10,13,12)(11,14,16,15)(18,23)(19,24)(25,29)(26,31);(2,6)(3,7)(9,12,13,10)(11,15,16,14)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32);(1,3,5,7)(2,4,6,8)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32);(1,4)(2,7)(3,6)(5,8)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=g,e^2=i,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7923835);