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 | |
4897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,12)(14,15)(17,18)(19,24)(20,21)(22,26)(23,27)(25,32)(28,31)(29,30);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14)(19,27)(23,24)(28,30)(29,31);(2,5)(3,7)(9,11,13,16)(10,14,12,15)(17,19,22,27)(18,23,26,24)(20,30,25,28)(21,31,32,29);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,20,22,25)(18,21,26,32)(19,28,27,30)(23,29,24,31);(1,3)(2,6)(4,7)(5,8)(17,19)(18,24)(20,30)(21,29)(22,27)(23,26)(25,28)(31,32);(1,2)(3,6)(4,5)(7,8)(17,21)(18,25)(19,29)(20,26)(22,32)(23,30)(24,28)(27,31)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8891038);