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 | |
4851 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)(3,5)(4,7)(6,8)(10,14)(15,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10,13,14)(11,15,12,16);(1,2)(3,6)(4,7)(5,8)(18,23)(19,24)(25,30)(26,31);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,16,14,15)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32);(1,2)(3,6)(4,7)(5,8)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)Polycyclic presentation:
2^9|b^2=g,d^2=gh,e^2=g,f^2=h,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, 7874486);