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 | #34911 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4597 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)(3,4)(5,7)(6,8)(9,10,12,14)(11,13,15,16)(21,29)(25,30)(26,31)(28,32);(1,3,6,5)(2,4,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,25)(22,23)(24,27)(26,28)(29,30)(31,32);(1,3,6,5)(2,7,8,4)(9,11,12,15)(10,13,14,16)(18,23)(20,27)(25,30)(28,32);(1,4,6,7)(2,3,8,5)(17,19,22,24)(18,20,23,27)(21,26,29,31)(25,28,30,32);(1,3,6,5)(2,4,8,7)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32);(1,5,6,3)(2,7,8,4)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,28,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=hi,e^2=h,f^2=hi,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7588953);