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 | #50640 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,21,16)(12,17,22,20)(13,23,24,18)(25,26)(27,32)(28,33)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2)(3,4)(5,6)(7,8)(10,15)(12,22)(16,19)(18,23)(25,27,31,33)(26,32,35,28)(29,36,34,40)(30,37,39,38);(2,6)(3,7)(10,15)(12,22)(16,19)(18,23)(25,28,31,32)(26,27,35,33)(29,38,34,37)(30,36,39,40);(3,7)(4,8)(9,11)(10,16)(12,23)(13,17)(14,21)(15,19)(18,22)(20,24)(25,29)(26,30)(27,36)(28,38)(31,34)(32,37)(33,40)(35,39);(1,3)(2,4)(5,7)(6,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,20)(16,24)(21,23)(25,28)(26,33)(27,35)(29,37)(30,36)(31,32)(34,38)(39,40);(1,4)(2,3)(5,8)(6,7)(9,13,14,24)(10,18,15,23)(11,20,21,17)(12,19,22,16)(25,30)(26,34)(27,37)(28,36)(29,35)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8971197);