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 | |
4955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,16)(13,18)(14,15)(19,22)(20,24)(21,23)(26,30)(31,33)(32,38)(39,40);(1,2)(3,7)(4,6)(5,8)(10,15)(11,18)(12,20)(22,23)(25,26,29,30)(27,31,35,33)(28,32,37,38)(34,39,36,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(16,24)(19,21);(2,6)(5,7)(9,11)(10,13)(12,21)(14,18)(15,17)(16,22)(19,20)(23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,23)(14,20)(15,24)(17,22)(18,21)(25,27)(26,31)(28,36)(29,35)(30,33)(32,40)(34,37)(38,39);(1,3)(2,7)(4,8)(5,6)(9,13)(10,11)(12,22)(14,17)(15,18)(16,21)(19,24)(20,23)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)Polycyclic presentation:
2^9|b^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10127073);