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 | |
4808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,14)(11,12,15,16)(17,18,23,20)(19,24,21,25)(22,28,30,26)(27,32,29,31);(1,2)(3,7)(4,5)(6,8)(10,14)(11,15)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32);(2,5)(3,8)(9,11)(10,12)(13,15)(14,16)(18,20)(19,21)(22,30)(31,32);(1,3,4,8)(2,6,5,7)(9,12)(10,11)(13,16)(14,15)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32);(1,3)(2,6)(4,8)(5,7)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31);(1,2)(3,6)(4,5)(7,8)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gi,e^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, 8969478);