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 | |
5229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30);(1,2)(3,5)(4,6)(7,8)(17,19)(18,23)(20,28)(21,22)(24,31)(25,26)(27,29)(30,32);(9,10,12,14)(11,13,16,15)(19,21)(23,26)(27,28)(30,31);(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,31)(28,30)(29,32);(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,32)(26,31);(1,3,4,7)(2,5,6,8)(17,21,22,19)(18,23,25,26)(20,28,29,27)(24,30,32,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7714000);