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 | |
5182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,13)(14,16)(18,21)(23,25)(24,29)(30,31);(1,2,7,8)(3,4,6,5)(10,13)(14,15)(20,28)(24,29)(26,32)(30,31);(1,3,7,6)(2,5,8,4)(9,10)(11,14)(12,13)(15,16)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,4,7,5)(2,3,8,6)(9,11)(10,14)(12,16)(13,15)(17,19,22,27)(18,23,21,25)(20,26,28,32)(24,30,29,31);(1,5,7,4)(2,3,8,6)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,6,7,3)(2,5,8,4)(17,21,22,18)(19,25,27,23)(20,29,28,24)(26,31,32,30)Polycyclic presentation:
2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9685672);