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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,12)(13,14)(15,16)(17,18)(19,24)(20,27)(21,29)(22,23)(25,31)(26,28)(30,32);(1,2,5,8)(3,4,7,6)(10,14)(11,15)(17,19,22,27)(18,20,23,24)(21,26,31,30)(25,28,29,32);(1,3)(2,6)(4,8)(5,7)(9,11,13,15)(10,12,14,16)(18,23)(20,24)(25,29)(28,32);(9,12)(10,15)(11,14)(13,16)(17,20,22,24)(18,19,23,27)(21,28,31,32)(25,26,29,30);(1,3)(2,4)(5,7)(6,8)(17,19)(18,24)(20,23)(21,30)(22,27)(25,28)(26,31)(29,32);(1,4)(2,7)(3,8)(5,6)(17,21)(18,25)(19,26)(20,28)(22,31)(23,29)(24,32)(27,30)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9582186);