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 | |
4837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(10,12)(14,15)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,31)(29,30);(3,7)(5,8)(9,10,13,12)(11,15,16,14)(17,19)(18,24)(20,27)(21,22)(23,28)(25,31)(26,29)(30,32);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,12,15)(18,25)(19,28)(21,32)(27,29);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,15,16,14)(17,20,23,26)(18,21,25,32)(19,29,28,27)(22,24,30,31);(1,2)(3,5)(4,6)(7,8)(17,21)(18,26)(19,30)(20,25)(22,28)(23,32)(24,27)(29,31);(17,22,23,30)(18,27,25,29)(19,21,28,32)(20,31,26,24)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=g,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541480);