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 | #37737 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,20)(12,21)(13,19)(14,24)(15,17)(16,22)(18,23)(27,30)(32,34)(35,37)(39,40);(2,6)(3,7)(9,11,15,22)(10,16,17,20)(12,13,23,24)(14,18,19,21)(25,26,31,28)(27,32,30,34)(29,36,38,33)(35,40,37,39);(1,3)(2,4)(5,7)(6,8)(10,17)(12,23)(13,24)(16,20)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,4)(2,3)(5,8)(6,7)(9,12)(10,18)(11,13)(14,20)(15,23)(16,19)(17,21)(22,24)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(9,13)(10,19)(11,12)(14,17)(15,24)(16,18)(20,21)(22,23)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(9,14,15,19)(10,13,17,24)(11,21,22,18)(12,20,23,16)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8714089);