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 | #57958 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,15)(11,16)(17,22)(18,23)(25,26,31,29)(27,32,30,37)(28,33,35,38)(34,39,36,40);(1,3,5,8)(2,6,7,4)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,27,31,30)(26,32,29,37)(28,34,35,36)(33,39,38,40);(1,3)(2,4)(5,8)(6,7)(11,16)(12,19)(13,21)(17,22)(26,29)(27,30)(28,35)(39,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28)(26,33)(27,34)(29,38)(30,36)(31,35)(32,39)(37,40);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29,31,26)(27,32,30,37)(28,33,35,38)(34,40,36,39);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30,31,27)(26,32,29,37)(28,36,35,34)(33,39,38,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8882566);