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 | |
4750 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)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32);(9,10)(11,14)(12,13)(15,16)(18,22)(23,25)(24,28)(30,31);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,18,21,22)(19,25,27,23)(20,24,29,28)(26,31,32,30);(1,3,5,7)(2,4,6,8)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,4)(2,7)(3,6)(5,8)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31);(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)Polycyclic presentation:
2^9|c^2=g,d^2=hi,e^2=i,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, 7622081);