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 | |
4332 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,26)(27,28)(29,33)(30,38)(31,32)(34,39)(35,40)(36,37);(2,5)(6,7)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(26,32)(28,33)(34,38)(36,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,29)(26,33,32,28)(30,35,39,37)(34,40,38,36);(1,3)(2,6)(4,8)(5,7)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,28,31,33)(26,27,32,29)(30,36,39,40)(34,35,38,37);(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,29,31,27)(26,33,32,28)(30,35,39,37)(34,36,38,40);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,38)(33,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9854635);