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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4622 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)(9,10,14,15)(11,17,18,16)(12,13,21,23)(19,20,22,24)(25,26,29,33)(27,30,36,39)(28,32,37,31)(34,40,35,38);(1,3,5,8)(2,6,7,4)(10,15)(12,21)(16,17)(19,22)(25,27,29,36)(26,30,33,39)(28,35,37,34)(31,40,32,38);(1,2)(3,4)(5,7)(6,8)(10,15)(11,18)(13,23)(19,22)(25,28,29,37)(26,31,33,32)(27,34,36,35)(30,38,39,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,21)(23,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,33)(34,36)(39,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,13)(11,19)(14,21)(15,23)(16,24)(17,20)(18,22)(25,27)(26,30)(28,34)(29,36)(31,38)(32,40)(33,39)(35,37);(1,4)(2,3)(5,6)(7,8)(9,13)(10,12)(11,20)(14,23)(15,21)(16,22)(17,19)(18,24)(25,26)(27,30)(28,32)(29,33)(31,37)(34,40)(35,38)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815583);