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 | |
5089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,14)(11,12,15,16)(17,18)(19,24)(20,26)(21,27)(22,31)(23,25)(28,30)(29,32);(1,2,5,6)(3,7,8,4)(10,14)(11,15)(17,19,23,22)(18,24,25,31)(20,28,29,27)(21,26,30,32);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,15)(14,16)(18,25)(19,22)(20,29)(21,30);(2,6)(3,8)(9,12)(10,11)(13,16)(14,15)(17,20,23,29)(18,26,25,32)(19,27,22,28)(21,31,30,24);(1,3)(2,4)(5,8)(6,7)(17,21)(18,27)(19,26)(20,24)(22,32)(23,30)(25,28)(29,31);(1,4)(2,3)(5,7)(6,8)(17,22,23,19)(18,24,25,31)(20,28,29,27)(21,32,30,26)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8968790);