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 | |
4477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(25,26)(27,35)(28,31)(29,38)(30,34)(32,37)(33,36)(39,40);(2,3)(6,7)(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,24,19)(25,27,31,37)(26,32,28,35)(29,30,36,40)(33,34,38,39);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(16,23)(18,24)(26,28)(32,35)(33,38)(34,39);(1,4,5,8)(2,6,3,7)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,28,31,26)(27,32,37,35)(29,38,36,33)(30,34,40,39);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29)(26,33)(27,30)(28,38)(31,36)(32,34)(35,39)(37,40);(9,13)(10,17)(11,15)(12,19)(14,22)(16,18)(20,21)(23,24)(25,30)(26,34)(27,36)(28,39)(29,37)(31,40)(32,38)(33,35)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8616365);