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 | #37444 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,5)(3,8,4,7)(10,13)(14,16)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32);(1,3,6,4)(2,7,5,8)(9,10,12,13)(11,14,15,16)(19,26)(23,30)(27,31)(28,32);(1,3,6,4)(2,7,5,8)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32);(1,3,6,4)(2,7,5,8);(1,4,6,3)(2,7,5,8)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32);(1,5,6,2)(3,8,4,7)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561763);