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 | |
4965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18,22,25)(19,23,28,32)(20,24,30,21)(26,27,29,31);(1,2,5,4)(3,6,8,7)(10,13)(14,15)(17,19,22,28)(18,23,25,32)(20,29,30,26)(21,27,24,31);(2,4)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,31)(22,28)(24,27)(25,32)(29,30);(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,26)(21,25)(22,30)(23,27)(28,29)(31,32);(1,3,5,8)(2,6,4,7);(1,4,5,2)(3,7,8,6)(17,21,22,24)(18,20,25,30)(19,27,28,31)(23,29,32,26)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,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, 8103156);