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 | |
5146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,7)(3,5,4,8)(11,15)(13,16)(21,29)(24,30)(26,31)(27,32);(2,7)(3,4)(9,10,12,14)(11,13,15,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32);(1,3,6,4)(2,8,7,5)(9,11)(10,13)(12,15)(14,16)(18,23)(19,25)(24,30)(26,31);(1,4,6,3)(2,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32);(1,5)(2,3)(4,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31);(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066542);