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 | |
5023 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)(11,15)(12,13)(14,16)(17,18)(19,26)(20,21)(22,24)(23,25)(27,31)(28,29)(30,32);(3,7)(5,8)(10,13)(14,15)(19,20)(21,26)(27,28)(29,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,21,25,26)(22,27,30,28)(24,29,32,31);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,16)(13,15)(17,20,23,19)(18,21,25,26)(22,27,30,28)(24,31,32,29);(1,2)(3,5)(4,6)(7,8)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,32)(30,31);(17,22,23,30)(18,24,25,32)(19,27,20,28)(21,29,26,31)Polycyclic presentation:
2^9|a^2=ghi,c^2=gi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8547026);