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 | #35223 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,7)(6,8)(10,13)(14,16)(21,29)(24,31)(27,30)(28,32);(1,3)(2,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31);(1,3)(2,6)(4,7)(5,8)(17,19,22,23)(18,20,25,26)(21,27,29,30)(24,28,31,32);(2,7)(4,6)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32);(1,4,5,6)(2,3,7,8)(9,11,12,15)(10,14,13,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,30)(27,31)(29,32);(1,3)(2,6)(4,7)(5,8)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|b^2=gh,c^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7552483);