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 | |
4889 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18)(19,27)(20,22)(21,24)(23,25)(26,30)(28,32)(29,31);(1,2)(3,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(17,19,23,21)(18,24,25,27)(20,28,30,31)(22,29,26,32);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,14,13,16)(18,25)(20,30)(24,27)(28,31);(1,2)(3,6)(4,7)(5,8)(17,20)(18,26)(19,28)(21,31)(22,25)(23,30)(24,32)(27,29);(1,2)(3,5)(4,7)(6,8)(17,21,23,19)(18,24,25,27)(20,31,30,28)(22,29,26,32);(1,3)(2,6)(4,8)(5,7)(17,22,23,26)(18,20,25,30)(19,29,21,32)(24,28,27,31)Polycyclic presentation:
2^9|b^2=g,c^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566797);