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 | |
5266 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)(3,7)(4,5)(6,8)(10,12)(14,16)(17,18)(19,22)(20,25)(21,26)(23,27)(24,30)(28,29)(31,32);(3,8)(6,7)(11,15)(14,16)(17,19,23,30)(18,24,27,22)(20,28,26,32)(21,29,25,31);(1,3,5,8)(2,6,4,7)(9,10,13,12)(11,14,15,16)(17,20,23,26)(18,25,27,21)(19,28,30,32)(22,29,24,31);(3,8)(6,7)(17,20)(18,21)(19,28)(22,31)(23,26)(24,29)(25,27)(30,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,21)(18,26)(19,29)(20,27)(22,28)(23,25)(24,32)(30,31);(1,4,5,2)(3,6,8,7)(9,12,13,10)(11,16,15,14)(17,22)(18,19)(20,31)(21,28)(23,24)(25,32)(26,29)(27,30)Polycyclic presentation:
2^9|b^2=g,c^2=gh,f^2=h,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, 8877758);