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 | |
4629 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)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(26,32)(28,37)(33,34)(35,38);(1,2,5,4)(3,6,7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,26)(27,33)(28,30)(29,34)(31,32)(35,39)(36,38)(37,40);(2,4)(6,8)(10,14)(15,17)(16,20)(23,24)(25,27,31,29)(26,33,32,34)(28,35,37,38)(30,39,40,36);(3,7)(6,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,28)(26,30)(27,35)(29,38)(31,37)(32,40)(33,39)(34,36);(1,3)(2,6)(4,8)(5,7)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,29,31,27)(26,34,32,33)(28,35,37,38)(30,39,40,36);(1,4,5,2)(3,8,7,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,30)(26,28)(27,36)(29,39)(31,40)(32,37)(33,38)(34,35)Polycyclic presentation:
2^9|b^2=hi,c^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8233098);