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 | |
5722 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)(11,16)(12,14)(13,15)(18,23)(19,25)(26,31)(27,32);(1,2)(3,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31);(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32);(1,2)(3,5)(4,7)(6,8)(9,12,13,11)(10,14,15,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)Polycyclic presentation:
2^9|b^2=g,d^2=h,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, 7560657);