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 | |
4505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,14)(12,15)(13,16)(17,18,24,28)(19,25,21,22)(20,29,31,26)(23,30,32,27);(2,5)(6,7)(9,10,13,16)(11,12,15,14)(17,19,24,21)(18,25,28,22)(20,30,31,27)(23,29,32,26);(1,2)(3,6)(4,5)(7,8)(17,20,24,31)(18,26,28,29)(19,27,21,30)(22,23,25,32);(9,11)(10,12)(13,15)(14,16)(17,21,24,19)(18,22,28,25)(20,27,31,30)(23,26,32,29);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,22)(18,19)(20,23)(21,28)(24,25)(26,27)(29,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,14)(10,15,16,11)(17,23)(18,27)(19,29)(20,25)(21,26)(22,31)(24,32)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9686667);