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 | |
4833 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)(9,10,13,15)(11,12,16,14);(3,4)(7,8)(9,11,13,16)(10,14,15,12)(17,18,23,27)(19,28,30,24)(20,22,26,32)(21,25,29,31);(1,3)(2,7)(4,6)(5,8)(9,11)(10,12)(13,16)(14,15)(17,19,23,30)(18,24,27,28)(20,21,26,29)(22,31,32,25);(3,4)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20,23,26)(18,22,27,32)(19,29,30,21)(24,25,28,31);(1,4,6,3)(2,7,5,8)(17,21,23,29)(18,25,27,31)(19,20,30,26)(22,24,32,28);(1,5,6,2)(3,7,4,8)(17,22)(18,26)(19,25)(20,27)(21,24)(23,32)(28,29)(30,31)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,d^2=ghi,e^2=gi,f^2=h,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, 8923849);