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 | |
4819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,4)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,23,21)(19,20,24,26)(22,25,31,29)(27,28,32,30);(1,2,5,8)(3,6,4,7)(10,13)(14,15)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,30)(31,32);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(19,24)(20,26)(27,32)(28,30);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,20,23,26)(18,24,21,19)(22,30,31,28)(25,27,29,32);(1,4,5,3)(2,7,8,6)(17,21,23,18)(19,26,24,20)(22,25,31,29)(27,28,32,30);(17,22,23,31)(18,25,21,29)(19,27,24,32)(20,28,26,30)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8633908);