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 | |
4916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,6,8,4)(17,18)(19,21)(20,28)(22,24)(23,26)(25,30)(27,32)(29,31);(1,3,5,8)(2,4,7,6)(9,10)(11,15)(12,13)(14,16)(17,19,22,26)(18,23,24,21)(20,29,30,27)(25,31,28,32);(1,3,5,8)(2,6,7,4)(10,13)(11,16)(18,24)(20,30)(21,23)(27,29);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,27)(21,31)(22,30)(23,32)(24,28)(26,29);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,28)(21,26)(22,24)(25,30)(27,31)(29,32);(1,4,5,6)(2,3,7,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,23)(18,26,24,19)(20,31,30,32)(25,29,28,27)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969420);