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 | |
4428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,16)(12,21,22,13)(19,20,24,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,2)(3,4)(5,6)(7,8)(10,15)(11,18)(12,22)(20,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(2,6)(4,7)(10,15)(13,21)(16,17)(20,23)(26,30)(31,33)(32,36)(38,39);(9,11,14,18)(10,16,15,17)(12,19,22,24)(13,20,21,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,13)(11,19)(14,22)(15,21)(16,20)(17,23)(18,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,4,5,7)(2,3,6,8)(9,13)(10,12)(11,20)(14,21)(15,22)(16,19)(17,24)(18,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=g,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8139482);