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 | |
4507 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)(25,26)(27,30)(28,29)(31,33)(32,34)(35,38)(36,39)(37,40);(3,7)(6,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27,31,34)(26,32,33,30)(28,35,37,39)(29,36,40,38);(1,2)(3,6)(4,5)(7,8)(10,14)(11,17)(16,19)(18,20)(26,33)(28,37)(30,32)(35,39);(9,11,13,17)(10,15,14,22)(12,20,21,18)(16,24,19,23)(25,28,31,37)(26,29,33,40)(27,35,34,39)(30,38,32,36);(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)(26,28)(27,36)(30,39)(31,40)(32,35)(33,37)(34,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30,31,32)(26,34,33,27)(28,38,37,36)(29,39,40,35)Polycyclic presentation:
2^9|b^2=g,d^2=gi,e^2=gh,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, 8837294);