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 | #39166 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4765 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)(3,4)(5,7)(6,8)(10,13)(14,16)(17,18)(19,27)(20,24)(21,31)(22,26)(23,29)(25,32)(28,30);(1,3,6,5)(2,4,8,7)(9,10,12,13)(11,14,15,16)(17,19,22,29)(18,23,26,27)(20,21,28,32)(24,25,30,31);(1,3,6,5)(2,7,8,4)(17,18,22,26)(19,27,29,23)(20,30,28,24)(21,25,32,31);(1,4,6,7)(2,3,8,5)(17,20,22,28)(18,24,26,30)(19,21,29,32)(23,25,27,31);(1,5,6,3)(2,7,8,4)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,30)(31,32);(17,21)(18,25)(19,28)(20,29)(22,32)(23,30)(24,27)(26,31)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7578164);