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 | #37834 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,7,8)(9,10)(11,18)(12,21)(13,22)(14,16)(15,17)(19,20)(23,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,3,6,7)(2,4,5,8)(9,11,15,14)(10,16,17,18)(12,19,22,24)(13,20,21,23)(26,31)(27,32)(33,38)(34,39);(1,4)(2,3)(5,7)(6,8)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(1,5,6,2)(3,4,7,8)(9,12)(10,13)(11,19)(14,24)(15,22)(16,20)(17,21)(18,23)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(9,13)(10,12)(11,20)(14,23)(15,21)(16,19)(17,22)(18,24);(1,4)(2,7)(3,5)(6,8)(9,14,15,11)(10,16,17,18)(12,19,22,24)(13,23,21,20)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=hi,d^2=h,f^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, 8343724);