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 | |
4828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18,23,25)(19,24,28,22)(20,26,29,32)(21,30,31,27);(1,2)(3,8)(4,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,28)(18,24,25,22)(20,21,29,31)(26,30,32,27);(2,7)(3,4)(9,11,12,15)(10,14,13,16)(18,25)(19,28)(21,31)(26,32);(1,3,6,4)(2,8,7,5)(17,20,23,29)(18,26,25,32)(19,21,28,31)(22,27,24,30);(1,4,6,3)(2,5,7,8)(17,21)(18,27)(19,29)(20,28)(22,32)(23,31)(24,26)(25,30);(1,5)(2,3)(4,7)(6,8)(17,22,23,24)(18,19,25,28)(20,27,29,30)(21,32,31,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644827);