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 | |
5017 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,21,25,23)(20,28,29,24)(27,30,32,31);(1,3,7,8)(2,6,5,4)(9,10,12,14)(11,13,16,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(1,4,7,6)(2,3,5,8)(9,10)(11,13)(12,14)(15,16)(20,29)(24,28)(27,32)(30,31);(3,8)(4,6)(9,11)(10,13)(12,16)(14,15)(17,20,22,29)(18,24,26,28)(19,27,25,32)(21,31,23,30);(1,5,7,2)(3,4,8,6)(17,19)(18,21)(20,27)(22,25)(23,26)(24,31)(28,30)(29,32);(1,6,7,4)(2,3,5,8)(17,21)(18,25)(19,26)(20,30)(22,23)(24,27)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8625641);