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 | #45257 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4620 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)(5,8)(6,7)(10,13)(14,16)(17,18,23,20)(19,24,21,25)(22,28,30,26)(27,32,29,31);(1,3,6,5)(2,4,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,25)(21,23)(22,27)(26,31)(28,32)(29,30);(2,7)(4,8)(18,20)(19,21)(26,28)(27,29);(1,4,6,8)(2,3,7,5)(9,11,12,15)(10,14,13,16)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31);(1,3,6,5)(2,4,7,8)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31);(1,5,6,3)(2,8,7,4)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,d^2=gh,e^2=h,f^2=hi,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, 7588534);