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) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,4,6,7)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,30)(21,31)(22,27)(23,24)(25,29)(26,32);(1,3)(2,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(18,24)(19,29)(22,30)(26,31);(1,4)(2,6)(3,8)(5,7)(17,19,23,29)(18,25,24,28)(20,31,27,26)(21,30,32,22);(1,3)(2,7)(4,8)(5,6)(17,20,23,27)(18,22,24,30)(19,26,29,31)(21,25,32,28);(1,3)(2,4)(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,21,23,32)(18,26,24,31)(19,22,29,30)(20,25,27,28);(1,4)(2,6)(3,8)(5,7)(17,22)(18,27)(19,21)(20,24)(23,30)(25,26)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875258);