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 |
| prime radical | #41268 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,21,22,16)(12,13,19,18)(14,24,23,20)(25,26,31,34)(27,29,32,28)(30,33,38,40)(35,37,39,36);(1,2,5,8)(3,4,7,6)(9,11,15,22)(10,16,17,21)(12,20,19,24)(13,23,18,14)(25,27,31,32)(26,28,34,29)(30,35,38,39)(33,36,40,37);(1,3)(2,6)(4,8)(5,7)(10,17)(12,19)(16,21)(20,24)(25,28,31,29)(26,27,34,32)(30,36,38,37)(33,35,40,39);(9,12,15,19)(10,18,17,13)(11,20,22,24)(14,21,23,16)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,18)(10,19,17,12)(11,14,22,23)(16,20,21,24)(25,29,31,28)(26,32,34,27)(30,36,38,37)(33,35,40,39);(1,4)(2,7)(3,8)(5,6)(9,14)(10,20)(11,13)(12,21)(15,23)(16,19)(17,24)(18,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9582202);