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 | #2430 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,7)(3,4,5,8)(9,10,12,14)(11,13,15,16)(17,18,22,27)(19,24,29,23)(20,25,30,32)(21,26,31,28);(1,3,6,5)(2,4,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,29)(18,23,27,24)(20,28,30,26)(21,25,31,32);(1,4)(2,3)(5,7)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,28)(21,25)(22,29)(23,27)(26,30)(31,32);(1,3,6,5)(2,4,7,8);(1,5,6,3)(2,4,7,8)(17,20)(18,25)(19,28)(21,24)(22,30)(23,31)(26,29)(27,32);(1,4)(2,5)(3,7)(6,8)(17,21)(18,26)(19,25)(20,23)(22,31)(24,30)(27,28)(29,32)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7771858);