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 | |
5124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(5,8)(10,12)(14,15)(17,18)(19,21)(20,25)(22,30)(23,24)(26,31)(27,32)(28,29);(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,15,16,14)(17,19,23,20)(18,21,24,25)(22,28,31,27)(26,32,30,29);(1,3)(2,5)(4,7)(6,8)(9,11,13,16)(10,14,12,15)(18,24)(19,20)(26,30)(27,28);(9,12,13,10)(11,15,16,14)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,32,30,29);(1,2)(3,5)(4,6)(7,8)(17,21,23,25)(18,20,24,19)(22,29,31,32)(26,28,30,27);(1,3)(2,5)(4,7)(6,8)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8769507);