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 | |
4963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,15)(11,14,16,12)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,15,12)(18,24)(19,20)(26,29)(27,28);(2,3)(6,7)(10,15)(11,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32);(4,8)(6,7)(11,16)(12,14)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30);(1,3,5,2)(4,7,8,6)(9,10)(11,12)(13,15)(14,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=g,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, 8520037);