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 | #35676 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,7)(3,5,4,8)(9,10,13,15)(11,16,12,14)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32);(1,3,6,4)(2,5,7,8)(9,11,13,12)(10,14,15,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32);(1,4,6,3)(2,5,7,8)(9,10)(11,14)(12,16)(13,15)(18,20)(19,21)(25,28)(27,29);(1,5)(2,4)(3,7)(6,8)(9,12,13,11)(10,14,15,16)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32);(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32);(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581790);