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 | #34976 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,5)(3,7,4,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,22)(21,25)(24,27)(26,31)(28,30)(29,32);(2,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,27)(18,23,20,24)(21,29,30,26)(25,32,28,31);(1,2,6,5)(3,8,4,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,32)(28,31)(29,30);(1,3,6,4)(2,7,5,8);(1,4,6,3)(2,8,5,7)(17,20,22,18)(19,24,27,23)(21,28,30,25)(26,32,29,31);(1,5,6,2)(3,8,4,7)(17,21,22,30)(18,25,20,28)(19,26,27,29)(23,31,24,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646530);