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 | |
5092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,21)(23,24)(25,32)(26,27)(28,30)(29,31);(1,2)(3,8)(4,6)(5,7)(10,13)(14,15)(17,19,23,25)(18,22,24,32)(20,28,29,26)(21,30,31,27);(1,3,5,6)(2,4,7,8)(9,10)(11,14)(12,13)(15,16)(18,24)(19,25)(20,29)(27,30);(1,2,5,7)(3,4,6,8)(9,11)(10,14)(12,16)(13,15)(17,20,23,29)(18,21,24,31)(19,26,25,28)(22,27,32,30);(1,2)(3,4)(5,7)(6,8)(17,21)(18,20)(19,27)(22,26)(23,31)(24,29)(25,30)(28,32);(1,4)(2,6)(3,7)(5,8)(17,22)(18,25)(19,24)(20,27)(21,28)(23,32)(26,31)(29,30)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=gi,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, 10157516);