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 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(25,26,30,29)(27,34,36,31)(28,32,37,33)(35,39,40,38);(1,2)(3,7)(4,8)(5,6)(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,24,19)(25,27)(26,31)(28,35)(29,34)(30,36)(32,38)(33,39)(37,40);(1,2)(3,7)(4,8)(5,6)(10,15)(11,17)(16,23)(18,24)(25,28,30,37)(26,32,29,33)(27,35,36,40)(31,38,34,39);(2,6)(4,7)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,28)(26,33)(27,35)(29,32)(30,37)(31,39)(34,38)(36,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,27)(26,34)(28,35)(29,31)(30,36)(32,39)(33,38)(37,40);(1,4,5,7)(2,8,6,3)(9,13)(10,17)(11,15)(12,19)(14,22)(16,18)(20,21)(23,24)(25,29,30,26)(27,34,36,31)(28,33,37,32)(35,39,40,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,f^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, 8616327);