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 | |
4290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(11,17)(15,22)(18,20)(23,24)(25,26)(27,35)(28,39)(29,33)(30,31)(32,34)(36,38)(37,40);(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27,32,29)(26,33,34,35)(28,36,30,37)(31,38,39,40);(1,3)(2,5)(4,7)(6,8)(10,14)(11,17)(16,19)(18,20)(25,28,32,30)(26,31,34,39)(27,36,29,37)(33,38,35,40);(9,11,13,17)(10,15,14,22)(12,20,21,18)(16,24,19,23)(25,29,32,27)(26,33,34,35)(28,36,30,37)(31,40,39,38);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,30,32,28)(26,31,34,39)(27,37,29,36)(33,38,35,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,31)(26,28)(27,38)(29,40)(30,34)(32,39)(33,36)(35,37)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837227);