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 | |
4588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,19,24)(12,17,21,22)(13,20,23,18)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(2,6)(4,7)(10,16)(11,19)(17,22)(18,20)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(9,12)(10,17)(11,18)(13,15)(14,21)(16,22)(19,20)(23,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,18)(11,17)(12,15)(14,23)(16,20)(19,22)(21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4,5,7)(2,3,6,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,24)(20,21)(22,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,f^2=hi,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, 8139511);