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 | |
5071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,15)(11,14,16,12)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,15,12)(18,19)(23,25)(24,26)(30,31);(2,3)(6,7)(10,15)(11,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31);(4,8)(6,7)(11,16)(12,14)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3,5,2)(4,7,8,6)(9,10)(11,12)(13,15)(14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8520038);