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 |
| prime radical | #2846 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,4,6,7)(10,13)(14,16)(17,18,23,25)(19,22,21,24)(20,28,29,26)(27,32,30,31);(1,3)(2,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,24,25,22)(20,27,29,30)(26,32,28,31);(1,4)(2,6)(3,8)(5,7)(18,25)(19,21)(20,29)(31,32);(1,3)(2,7)(4,8)(5,6)(9,11,12,15)(10,14,13,16)(17,20,23,29)(18,26,25,28)(19,27,21,30)(22,31,24,32);(1,3)(2,4)(5,6)(7,8)(17,21,23,19)(18,22,25,24)(20,30,29,27)(26,31,28,32);(1,4)(2,6)(3,8)(5,7)(17,22)(18,21)(19,25)(20,31)(23,24)(26,30)(27,28)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875461);