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 | |
4708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,25)(20,22)(21,28)(23,27)(24,30)(26,31)(29,32);(3,7)(5,8)(10,14)(11,15)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,12,14,16)(18,20)(23,25)(24,28)(31,32);(9,12)(10,15)(11,14)(13,16)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32);(1,3)(2,5)(4,7)(6,8)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32);(1,2)(3,5)(4,6)(7,8)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=i,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8780903);