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 | |
5924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,7)(3,5,4,8)(10,13)(14,16)(17,18)(19,27)(20,29)(21,26)(22,23)(24,28)(25,30)(31,32);(2,7)(3,4)(11,15)(14,16)(18,23)(19,28)(25,29)(26,32);(1,3,6,4)(2,8,7,5)(9,10)(11,14)(12,13)(15,16)(18,23)(20,30)(24,27)(26,32);(1,4,6,3)(2,5,7,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,28)(18,24,23,27)(20,26,30,32)(21,29,31,25);(1,5)(2,3)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,26)(21,24)(22,30)(23,29)(27,31)(28,32);(17,21)(18,26)(19,25)(20,24)(22,31)(23,32)(27,30)(28,29)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8434685);