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 | |
5126 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)(11,14)(12,16)(13,15)(17,18,22,23)(19,20,25,24)(21,29,30,26)(27,31,32,28);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,15,12)(18,23)(19,25)(26,29)(27,32);(9,11)(10,12)(13,16)(14,15)(18,23)(20,24)(26,29)(28,31);(9,12)(10,11)(13,14)(15,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28);(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,3,4,7)(2,5,6,8)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551545);