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 | |
4587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,18,23,26)(19,24,27,21)(20,22,25,32)(28,31,30,29);(1,2,4,6)(3,5,7,8)(10,14)(11,15)(17,19)(18,24)(20,30)(21,26)(22,29)(23,27)(25,28)(31,32);(1,3)(2,5)(4,7)(6,8)(9,11,13,15)(10,12,14,16)(19,27)(21,24)(28,30)(29,31);(9,12)(10,15)(11,14)(13,16)(17,20)(18,22)(19,28)(21,29)(23,25)(24,31)(26,32)(27,30);(1,2)(3,5)(4,6)(7,8)(17,21,23,24)(18,19,26,27)(20,31,25,29)(22,30,32,28);(1,3)(2,5)(4,7)(6,8)(17,22)(18,25)(19,29)(20,26)(21,30)(23,32)(24,28)(27,31)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9009108);