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 | |
5026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10,13,14)(11,12,15,16)(17,18)(19,21)(20,27)(22,23)(24,28)(25,31)(26,32)(29,30);(3,7)(6,8)(10,14)(11,15)(18,23)(19,25)(20,28)(30,32);(1,2,5,4)(3,6,7,8)(9,11)(10,12)(13,15)(14,16)(19,25)(21,31)(26,29)(30,32);(9,12)(10,11)(13,16)(14,15)(17,19)(18,21)(20,29)(22,25)(23,31)(24,32)(26,28)(27,30);(1,3)(2,6)(4,8)(5,7)(17,20)(18,24)(19,26)(21,30)(22,28)(23,27)(25,29)(31,32);(1,4,5,2)(3,8,7,6)(17,21)(18,19)(20,30)(22,31)(23,25)(24,26)(27,29)(28,32)Polycyclic presentation:
2^9|a^2=i,c^2=h,f^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,d^c=dgi,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, 8841550);