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 | |
4965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,26)(21,31)(22,32)(23,25)(27,30)(28,29);(1,3,5,8)(2,6,4,7)(17,19)(18,24)(20,23)(21,27)(22,28)(25,26)(29,32)(30,31);(10,13)(14,15)(18,25)(19,20)(21,32)(28,30);(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,26,25,24)(21,29,32,27)(22,30,31,28);(9,11)(10,14)(12,16)(13,15)(17,21)(18,22)(19,27)(20,29)(23,32)(24,28)(25,31)(26,30);(1,4,5,2)(3,7,8,6)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621070);