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 | |
5096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18,22,26)(19,21,25,23)(20,24,29,31)(27,30,32,28);(1,2)(3,8)(4,6)(5,7)(11,15)(14,16)(17,19,22,25)(18,23,26,21)(20,27,29,32)(24,28,31,30);(1,3,5,6)(2,4,7,8)(9,10)(11,14)(12,13)(15,16);(1,4,5,8)(2,3,7,6)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,28)(29,32)(30,31);(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,28)(25,32)(26,31);(1,4)(2,6)(3,7)(5,8)(17,21,22,23)(18,25,26,19)(20,28,29,30)(24,27,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8604516);