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 | |
4866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,14)(13,16)(18,22)(20,28)(23,25)(26,29);(1,3)(2,6)(4,8)(5,7)(9,10,12,14)(11,13,16,15)(18,22)(23,25)(24,30)(31,32);(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32);(1,2,5,8)(3,4,7,6)(9,11,12,16)(10,13,14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31);(1,3)(2,4)(5,7)(6,8)(17,19,21,27)(18,23,22,25)(20,29,28,26)(24,32,30,31);(1,4)(2,7)(3,8)(5,6)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10371783);