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 | |
5374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(17,18)(19,26)(20,28)(21,23)(22,25)(24,30)(27,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15);(1,2,5,4)(3,7,8,6)(9,11,13,14)(10,12,15,16)(17,19,22,21)(18,23,25,26)(20,27,30,29)(24,31,28,32);(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,12)(13,15)(14,16)(17,21,22,19)(18,23,25,26)(20,29,30,27)(24,31,28,32);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=g,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561801);