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 | |
4803 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)(19,28)(20,31)(21,22)(23,24)(25,29)(26,30)(27,32);(1,2,5,4)(3,7,8,6)(10,13)(14,15)(18,24)(19,29)(21,27)(26,31);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,27)(21,30)(22,31)(23,29)(24,28)(26,32);(3,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,20,23,30)(18,26,24,31)(19,27,29,21)(22,25,32,28);(1,3)(2,7)(4,6)(5,8)(17,21,23,27)(18,22,24,32)(19,30,29,20)(25,31,28,26);(1,4,5,2)(3,6,8,7)(17,22)(18,27)(19,31)(20,25)(21,24)(23,32)(26,29)(28,30)Polycyclic presentation:
2^9|b^2=gi,c^2=gi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8935782);