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 | #35784 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4325 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)(3,7)(4,5)(6,8)(9,10,15,18)(11,16,19,14)(12,13,17,23)(20,21,24,22)(25,26)(27,32)(28,31)(29,33)(30,34)(35,37)(36,39)(38,40);(3,8)(6,7)(9,11,15,19)(10,16,18,14)(12,20,17,24)(13,21,23,22)(25,27)(26,28)(29,35)(30,31)(32,34)(33,36)(37,40)(38,39);(1,3,5,8)(2,6,4,7)(11,19)(12,17)(13,23)(14,16)(25,28,30,32)(26,31,34,27)(29,37,38,36)(33,35,40,39);(3,8)(6,7)(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,23)(19,24)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,38)(39,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,17)(11,21)(12,18)(14,20)(15,23)(16,24)(19,22)(25,27)(26,32)(28,34)(29,35)(30,31)(33,37)(36,40)(38,39);(1,4,5,2)(3,6,8,7)(9,14)(10,11)(12,22)(13,20)(15,16)(17,21)(18,19)(23,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gh,f^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, 8889312);