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 | |
4657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,8)(4,5)(6,7)(9,10,14,15)(11,13,12,16)(18,22)(23,29)(24,30)(31,32);(1,3,6,5)(2,4,7,8)(10,15)(11,12)(18,22)(19,25)(24,30)(26,27);(2,7)(4,8)(9,11,14,12)(10,16,15,13)(17,18,21,22)(19,23,25,29)(20,24,28,30)(26,31,27,32);(2,7)(3,5)(9,12,14,11)(10,13,15,16)(17,19)(18,23)(20,27)(21,25)(22,29)(24,32)(26,28)(30,31);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30);(1,5,6,3)(2,4,7,8)(17,20,21,28)(18,24,22,30)(19,26,25,27)(23,31,29,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=i,d^2=ghi,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, 8557590);