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 | |
5292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,12)(13,16)(14,15)(18,23)(19,26)(25,32)(27,28);(3,8)(5,6)(9,11)(10,14)(12,16)(13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,16)(13,14);(3,8)(5,6)(10,16)(12,14)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,31,32,29);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,14)(15,16)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29);(1,2)(3,5)(4,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,32,28)Polycyclic presentation:
2^9|a^2=i,d^2=ghi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123679);