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 | |
4251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,8)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,23)(22,24)(25,26,31,29)(27,32,37,38)(28,34,30,33)(35,40,36,39);(2,7)(3,8)(10,15)(12,21)(16,23)(18,22)(25,27)(26,32)(28,35)(29,38)(30,36)(31,37)(33,39)(34,40);(1,3,6,8)(2,4,7,5)(12,21)(13,24)(17,19)(18,22)(25,28,31,30)(26,33,29,34)(27,35,37,36)(32,39,38,40);(1,4,6,5)(2,3,7,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,23)(17,24)(18,21);(1,2)(3,4)(5,8)(6,7)(9,12,14,21)(10,17,15,19)(11,18,20,22)(13,23,24,16)(25,29,31,26)(27,32,37,38)(28,34,30,33)(35,39,36,40);(1,5,6,4)(2,8,7,3)(9,13)(10,18)(11,19)(12,23)(14,24)(15,22)(16,21)(17,20)(25,30,31,28)(26,34,29,33)(27,36,37,35)(32,40,38,39)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8946350);