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 | |
4905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18,22,21)(19,25,26,23)(20,24,28,30)(27,32,29,31);(1,2,5,6)(3,4,7,8)(18,21)(19,26)(24,30)(27,29);(2,6)(3,7)(9,10)(11,14)(12,13)(15,16)(20,28)(24,30)(27,29)(31,32);(2,6)(3,7)(11,15)(14,16)(17,19,22,26)(18,23,21,25)(20,29,28,27)(24,32,30,31);(1,3)(2,4)(5,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20,22,28)(18,24,21,30)(19,27,26,29)(23,31,25,32);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,21,22,18)(19,25,26,23)(20,30,28,24)(27,32,29,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10236486);