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 | #52031 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,13)(14,15)(16,19)(17,24)(20,23)(21,22)(25,26)(27,32)(28,35)(29,33)(30,31)(34,37)(36,40)(38,39);(1,2,5,3)(4,6,8,7)(10,15)(11,19)(12,17)(21,23)(26,31)(28,32)(33,39)(36,37);(2,3)(6,7)(10,15)(12,17)(16,18)(20,22)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,37,39,36);(2,3)(4,8)(9,11,14,19)(10,16,15,18)(12,22,17,20)(13,23,24,21)(25,28)(26,27)(29,37)(30,32)(31,35)(33,40)(34,39)(36,38);(1,3,5,2)(4,7,8,6)(9,12,14,17)(10,13,15,24)(11,20,19,22)(16,21,18,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,37)(36,39)(38,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,21)(12,15)(14,24)(16,22)(18,20)(19,23)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,37)(35,40)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012489);