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 | |
4668 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)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26)(27,35)(28,33)(29,36)(30,32)(31,38)(34,40)(37,39);(1,2,5,7)(3,4,8,6)(10,11)(13,21)(15,17)(20,24)(25,27,30,38)(26,31,32,35)(28,36,39,34)(29,33,40,37);(1,3,5,8)(2,6,7,4)(12,19)(15,17)(20,24)(22,23)(26,32)(28,39)(31,35)(34,36);(1,3,5,8)(2,6,7,4)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,39)(26,33,32,37)(27,36,38,34)(29,31,40,35);(1,4)(2,3)(5,6)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,40)(26,34,32,36)(27,37,38,33)(28,31,39,35)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275526);