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 | |
5136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,15,16)(11,13,14,12)(19,25)(20,28)(23,26)(24,29);(1,2,5,4)(3,7,8,6)(9,11,15,14)(10,12,16,13)(17,18,22,21)(19,26,25,23)(20,24,28,29)(27,32,31,30);(2,4)(6,7)(9,12,15,13)(10,11,16,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,13,15,12)(10,11,16,14);(17,20,22,28)(18,24,21,29)(19,27,25,31)(23,30,26,32);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,12,16,13)(17,21,22,18)(19,23,25,26)(20,29,28,24)(27,30,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8102899);