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 | |
4917 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)(3,5)(4,6)(7,8)(10,13)(14,16)(18,23)(20,25)(24,31)(28,32);(9,10,12,13)(11,14,15,16)(17,18,22,23)(19,25,27,20)(21,24,30,31)(26,32,29,28);(1,2,4,6)(3,5,7,8)(18,23)(20,25)(24,31)(28,32);(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,19,22,27)(18,20,23,25)(21,29,30,26)(24,32,31,28);(1,3,4,7)(2,5,6,8)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32);(1,2)(3,5)(4,6)(7,8)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7985517);