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 | #34589 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,14,15)(11,17,12,16)(13,21,22,18)(19,23,20,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,3,5,4)(2,6,7,8)(10,15)(16,17)(18,21)(23,24)(26,31)(27,32)(33,38)(34,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(2,7)(6,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,23)(21,22)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,2)(3,6)(4,8)(5,7)(9,12,14,11)(10,17,15,16)(13,20,22,19)(18,24,21,23);(1,4,5,3)(2,8,7,6)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066475);