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 | |
4668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,29)(31,32);(1,2,5,4)(3,7,8,6)(10,13)(14,15)(18,23)(19,26)(25,28)(27,32);(2,4)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29);(9,10)(11,14)(12,13)(15,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,16)(13,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27);(1,4,5,2)(3,7,8,6)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=gh,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161903);