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 | |
4818 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,21)(22,24)(23,30)(25,27)(28,29)(31,32);(1,3,5,7)(2,4,6,8)(10,13)(14,15)(17,19,22,30)(18,23,24,26)(20,31,27,28)(21,29,25,32);(9,10)(11,14)(12,13)(15,16)(18,24)(20,27)(23,26)(28,31);(2,6)(3,7)(9,11,12,16)(10,14,13,15)(17,20,22,27)(18,25,24,21)(19,28,30,31)(23,29,26,32);(1,2)(3,4)(5,6)(7,8)(17,19)(18,26)(20,31)(21,32)(22,30)(23,24)(25,29)(27,28);(1,4)(2,7)(3,6)(5,8)(17,21,22,25)(18,27,24,20)(19,29,30,32)(23,28,26,31)Polycyclic presentation:
2^9|b^2=gi,d^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154844);