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 | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18,23,24)(19,21,28,25)(20,26,31,29)(22,30,32,27);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(18,24)(19,28)(26,29)(27,30);(2,5)(6,8)(17,19,23,28)(18,25,24,21)(20,30,31,27)(22,29,32,26);(17,20,23,31)(18,26,24,29)(19,27,28,30)(21,22,25,32);(1,2)(3,6)(4,5)(7,8)(17,21,23,25)(18,19,24,28)(20,22,31,32)(26,27,29,30);(1,3)(2,6)(4,7)(5,8)(17,22)(18,27)(19,29)(20,25)(21,31)(23,32)(24,30)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,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, 7615987);