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 | |
5071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,4)(3,7,5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31);(1,3,6,5)(2,7,4,8)(10,13)(14,15)(18,22)(23,25)(24,28)(31,32);(1,3,6,5)(2,8,4,7)(9,10)(11,14)(12,13)(15,16)(18,22)(23,25)(24,28)(31,32);(1,4,6,2)(3,7,5,8)(9,11,12,16)(10,14,13,15)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31);(1,5,6,3)(2,7,4,8)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31);(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657485);