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 | |
4418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,20)(13,17)(14,18)(15,21)(19,23)(22,24)(25,26)(27,30)(28,33)(29,34)(31,32)(35,40)(36,38)(37,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,19)(14,17)(15,23)(16,18)(20,24)(21,22)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,38,39,34);(1,3)(2,5)(4,8)(6,7)(12,21)(15,20)(19,22)(23,24)(26,32)(27,35)(28,37)(34,38);(3,8)(5,6)(9,12)(10,15)(11,19)(13,23)(14,21)(16,24)(17,22)(18,20)(25,28)(26,33)(27,36)(29,35)(30,38)(31,37)(32,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,22)(13,18)(14,17)(15,23)(19,21)(20,24)(25,29,31,36)(26,34,32,38)(27,37,35,28)(30,39,40,33);(1,2)(3,5)(4,7)(6,8)(9,13,14,16)(10,17,18,11)(12,23,21,24)(15,22,20,19)(25,30)(26,35)(27,32)(28,38)(29,33)(31,40)(34,37)(36,39)Polycyclic presentation:
2^9|b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9030826);