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 | #59167 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4345 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,23)(13,24)(14,18)(15,22)(16,21)(17,20)(26,32)(27,35)(28,37)(34,39);(1,3)(2,7)(4,8)(5,6)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,19)(23,24)(25,26,31,32)(27,30,35,40)(28,38,37,33)(29,34,36,39);(1,4)(2,3)(5,8)(6,7)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,34,38,39);(1,5,6,3)(2,4,8,7)(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,17,19,24)(25,28)(26,33)(27,36)(29,35)(30,34)(31,37)(32,38)(39,40);(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,16,19,23)(25,29,31,36)(26,34,32,39)(27,28,35,37)(30,38,40,33);(1,4)(2,5)(3,8)(6,7)(9,10)(11,15)(12,23)(13,17)(14,18)(16,21)(19,22)(20,24)(25,30)(26,35)(27,32)(28,39)(29,38)(31,40)(33,36)(34,37)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343764);