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 | |
5084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,21)(22,23)(24,27)(25,30)(28,29)(31,32);(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,16,15)(18,23)(19,27)(20,30)(29,32);(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(18,23)(21,25)(24,26)(29,32);(3,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,27)(18,24,23,26)(20,31,30,28)(21,29,25,32);(1,3)(2,6)(4,8)(5,7)(17,20)(18,25)(19,28)(21,23)(22,30)(24,29)(26,32)(27,31);(1,2)(3,6)(4,5)(7,8)(17,21,22,25)(18,20,23,30)(19,29,27,32)(24,31,26,28)Polycyclic presentation:
2^9|b^2=gh,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8255429);