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 | |
4894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(3,4)(10,13)(14,16)(18,22)(19,25)(24,27)(26,31);(1,2,6,5)(3,8,4,7)(11,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16);(1,4,6,3)(2,8,5,7)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(1,5,6,2)(3,8,4,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32);(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343441);