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 | |
4530 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)(27,34)(28,31)(29,33)(30,32)(35,40)(36,38)(37,39);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,19)(14,17)(15,23)(16,18)(20,24)(21,22)(25,27,31,30)(26,32,28,34)(29,37,38,35)(33,40,36,39);(1,3)(2,4)(5,6)(7,8)(12,21)(15,20)(19,22)(23,24)(26,28)(32,34)(33,36)(39,40);(3,6)(4,8)(9,12,14,21)(10,15,18,20)(11,19,17,22)(13,23,16,24)(25,28,31,26)(27,34,30,32)(29,33,38,36)(35,39,37,40);(1,4,5,8)(2,6,7,3)(9,11)(10,16)(12,22)(13,18)(14,17)(15,23)(19,21)(20,24)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(1,2)(3,4)(5,7)(6,8)(9,13,14,16)(10,17,18,11)(12,23,21,24)(15,22,20,19)(25,30,31,27)(26,34,28,32)(29,35,38,37)(33,39,36,40)Polycyclic presentation:
2^9|b^2=h,d^2=g,e^2=hi,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, 9068358);