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 | |
5135 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)(3,7)(10,13)(14,16)(17,18,22,24)(19,25,29,23)(20,21,26,31)(27,32,30,28);(1,2,5,6)(3,4,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,25)(26,30)(31,32);(2,6)(3,7)(18,24)(21,31)(23,25)(28,32);(3,7)(4,8)(17,19)(18,25)(20,30)(21,28)(22,29)(23,24)(26,27)(31,32);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16)(17,20,22,26)(18,21,24,31)(19,27,29,30)(23,28,25,32);(1,4)(2,7)(3,6)(5,8)(17,21)(18,26)(19,28)(20,24)(22,31)(23,30)(25,27)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7976910);