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 | |
4874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(4,8)(10,14)(15,16)(17,18)(19,24)(20,21)(22,31)(23,25)(26,30)(27,28)(29,32);(1,2,5,6)(3,4,7,8)(9,10,13,14)(11,15,12,16)(17,19,23,29)(18,24,25,32)(20,28,30,22)(21,27,26,31);(2,6)(3,7)(18,25)(21,26)(24,32)(27,31);(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,28)(21,25)(22,29)(23,30)(24,31)(27,32);(1,3,5,7)(2,4,6,8)(9,12,13,11)(10,16,14,15)(17,21,23,26)(18,20,25,30)(19,27,29,31)(22,24,28,32);(1,4)(2,7)(3,6)(5,8)(17,22,23,28)(18,27,25,31)(19,20,29,30)(21,24,26,32)Polycyclic presentation:
2^9|b^2=gh,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550078);