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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,4)(3,8,5,7)(10,14)(15,16)(19,25)(20,27)(23,29)(24,28);(9,10,13,14)(11,15,12,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(1,3,6,5)(2,7,4,8)(18,22)(23,29)(24,28)(30,32);(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,31)(28,32);(1,4,6,2)(3,7,5,8)(9,11,13,12)(10,15,14,16)(17,18)(19,23)(20,28)(21,22)(24,27)(25,29)(26,32)(30,31);(1,5,6,3)(2,8,4,7)(9,12,13,11)(10,16,14,15)(17,20)(18,24)(19,26)(21,27)(22,28)(23,30)(25,31)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7572051);