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 | |
4763 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,28,29,24)(20,32,26,22)(21,25,31,30);(1,3,5,6)(2,4,7,8)(9,10,12,14)(11,13,16,15)(19,29)(21,31)(24,28)(25,30);(9,10)(11,13)(12,14)(15,16)(17,19,23,29)(18,24,27,28)(20,25,26,30)(21,22,31,32);(9,11)(10,13)(12,16)(14,15)(17,20,23,26)(18,22,27,32)(19,30,29,25)(21,28,31,24);(1,2)(3,4)(5,7)(6,8)(17,21)(18,25)(19,22)(20,28)(23,31)(24,26)(27,30)(29,32);(1,4)(2,6)(3,7)(5,8)(17,22)(18,26)(19,31)(20,27)(21,29)(23,32)(24,30)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=di,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, 9620013);