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 |
| prime radical | #35436 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,8,7,4)(9,10,12,13)(11,14,15,16)(17,18)(19,26)(20,22)(21,24)(23,25)(27,32)(28,29)(30,31);(1,2,5,6)(3,4,7,8)(10,13)(14,16)(17,19,23,21)(18,24,25,26)(20,30,27,28)(22,29,32,31);(2,6)(4,8)(11,15)(14,16)(18,25)(22,32)(24,26)(29,31);(2,6)(3,7)(9,10,12,13)(11,14,15,16)(17,20,23,27)(18,22,25,32)(19,28,21,30)(24,31,26,29);(1,3,5,7)(2,4,6,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,19)(18,26,25,24)(20,28,27,30)(22,31,32,29);(1,4)(2,7)(3,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,22)(18,27)(19,29)(20,25)(21,31)(23,32)(24,28)(26,30)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840030);