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 | |
4636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,15)(11,17)(16,22)(18,23)(25,26)(27,31)(28,35)(29,32)(30,33)(34,39)(36,40)(37,38);(3,7)(5,8)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,2)(3,5)(4,6)(7,8)(12,19)(16,22)(18,23)(20,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8438854);