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 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(11,17)(15,22)(18,20)(23,24)(26,30)(27,35)(28,38)(34,40);(3,7)(6,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,26,31,30)(27,36,35,32)(28,33,38,39)(29,40,37,34);(1,2)(3,6)(4,5)(7,8)(10,14)(11,17)(16,19)(18,20)(25,27,31,35)(26,32,30,36)(28,37,38,29)(33,40,39,34);(9,11,13,17)(10,15,14,22)(12,20,21,18)(16,24,19,23)(25,28)(26,33)(27,37)(29,35)(30,39)(31,38)(32,40)(34,36);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,29,31,37)(26,34,30,40)(27,28,35,38)(32,33,36,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30,31,26)(27,36,35,32)(28,39,38,33)(29,40,37,34)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gi,e^2=h,f^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, 8837282);