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 |
| prime radical | #37660 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,4,8,6)(17,18)(19,24)(20,21)(22,27)(23,25)(26,31)(28,32)(29,30);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,19,23,28)(18,24,25,32)(20,29,26,27)(21,30,31,22);(2,7)(4,6)(9,10,12,13)(11,14,15,16)(18,25)(19,28)(21,31)(27,29);(2,7)(3,8)(11,15)(14,16)(17,20,23,26)(18,21,25,31)(19,27,28,29)(22,32,30,24);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,26)(19,22)(20,25)(23,31)(24,29)(27,32)(28,30);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,22)(18,27)(19,21)(20,24)(23,30)(25,29)(26,32)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gh,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, 8506526);