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 | |
4378 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)(10,15)(11,17)(16,23)(18,20)(25,26)(27,32)(28,34)(29,30)(31,36)(33,37)(35,40)(38,39);(2,3)(6,7)(9,10)(11,13)(12,16)(14,15)(17,22)(18,21)(19,23)(20,24)(25,27,31,38)(26,32,36,39)(28,35,37,29)(30,34,40,33);(1,3,5,2)(4,7,8,6)(11,17)(12,19)(13,22)(16,23)(25,28,31,37)(26,33,36,34)(27,29,38,35)(30,32,40,39);(1,4,5,8)(2,6,3,7)(9,11)(10,13)(12,20)(14,17)(15,22)(16,24)(18,19)(21,23)(25,28)(26,34)(27,35)(29,38)(30,39)(31,37)(32,40)(33,36);(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,29,31,35)(26,30,36,40)(27,37,38,28)(32,33,39,34);(9,13)(10,11)(12,21)(14,22)(15,17)(16,18)(19,24)(20,23)(25,30)(26,35)(27,34)(28,39)(29,36)(31,40)(32,37)(33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8900128);