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 | |
4668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(3,5)(10,13)(14,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(1,2,6,4)(3,7,5,8)(11,15)(14,16)(18,19)(23,25)(24,26)(30,31);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3,6,5)(2,8,4,7)(9,11)(10,14)(12,15)(13,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4,6,2)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31);(1,5,6,3)(2,8,4,7)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8059403);