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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(10,11)(14,15)(17,18)(19,27)(20,24)(21,22)(23,25)(26,32)(28,29)(30,31);(9,10,13,11)(12,14,16,15)(17,19,23,20)(18,24,25,27)(21,30,26,28)(22,29,32,31);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,14,16,15)(18,25)(19,20)(21,26)(29,31);(1,4,5,2)(3,6,8,7)(9,12,13,16)(10,14,11,15)(17,20,23,19)(18,24,25,27)(21,30,26,28)(22,31,32,29);(17,21,23,26)(18,22,25,32)(19,28,20,30)(24,31,27,29);(17,22)(18,26)(19,29)(20,31)(21,25)(23,32)(24,28)(27,30)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=hi,d^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7585201);