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 | #2656 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,7)(3,8)(10,13)(14,16)(18,23)(21,32)(24,30)(25,28);(1,2)(3,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,24,27,30)(20,28,31,25)(21,26,32,29);(1,3,6,8)(2,4,7,5)(18,23)(19,27)(25,28)(26,29);(1,4,6,5)(2,3,7,8)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31);(1,5,6,4)(2,8,7,3)(9,11,12,15)(10,14,13,16)(17,20,22,31)(18,25,23,28)(19,26,27,29)(21,30,32,24);(17,21)(18,26)(19,28)(20,30)(22,32)(23,29)(24,31)(25,27)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538277);