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 | #39335 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,7,5,4)(10,13)(14,16)(17,18,22,23)(19,21,26,24)(20,25,30,31)(27,29,28,32);(1,3,6,5)(2,7,8,4)(9,10,12,13)(11,14,15,16)(18,23)(19,26)(25,31)(27,28);(2,8)(4,7)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,29,31,32);(1,4)(2,3)(5,8)(6,7)(17,19)(18,21)(20,27)(22,26)(23,24)(25,29)(28,30)(31,32);(17,20,22,30)(18,25,23,31)(19,27,26,28)(21,29,24,32);(1,5,6,3)(2,4,8,7)(9,11,12,15)(10,14,13,16)(17,21)(18,19)(20,29)(22,24)(23,26)(25,27)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581516);