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 | #59758 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4617 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)(9,10,13,16)(11,14,12,15)(18,23)(21,25)(24,30)(28,31);(1,3,6,5)(2,4,8,7)(9,10)(11,14)(12,15)(13,16)(17,18,22,23)(19,24,29,30)(20,21,26,25)(27,28,32,31);(1,3,6,5)(2,7,8,4)(18,23)(20,26)(24,30)(27,32);(1,4,6,7)(2,3,8,5)(9,11,13,12)(10,14,16,15)(17,19,22,29)(18,24,23,30)(20,27,26,32)(21,28,25,31);(1,3,6,5)(2,4,8,7)(9,12,13,11)(10,15,16,14)(17,20,22,26)(18,25,23,21)(19,27,29,32)(24,31,30,28);(1,5,6,3)(2,7,8,4)(17,21,22,25)(18,26,23,20)(19,28,29,31)(24,32,30,27)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,d^2=gh,e^2=gh,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, 7588550);