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 | #45332 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,4)(3,7,5,8)(10,14)(15,16)(18,24)(22,32)(25,28)(26,30);(1,3,6,5)(2,7,4,8)(9,10,13,14)(11,15,12,16)(17,18)(19,28)(20,25)(21,30)(22,27)(23,24)(26,31)(29,32);(2,4)(7,8)(17,19,23,20)(18,25,24,28)(21,27,31,29)(22,30,32,26);(1,4,6,2)(3,8,5,7)(9,11,13,12)(10,15,14,16)(17,20,23,19)(18,25,24,28)(21,27,31,29)(22,26,32,30);(17,21,23,31)(18,26,24,30)(19,29,20,27)(22,25,32,28);(1,5,6,3)(2,8,4,7)(9,12,13,11)(10,16,14,15)(17,22,23,32)(18,27,24,29)(19,30,20,26)(21,25,31,28)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,d^2=gh,e^2=i,f^2=gh,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, 7585801);