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 | |
4709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(10,12)(16,19)(17,20)(23,24)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(2,6)(3,8)(9,10,15,12)(11,16,14,19)(13,17,22,20)(18,23,21,24)(25,26,31,28)(27,32,30,34)(29,36,38,33)(35,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,11,15,14)(10,16,12,19)(13,21,22,18)(17,24,20,23)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,4)(2,3)(5,7)(6,8)(9,12,15,10)(11,16,14,19)(13,20,22,17)(18,23,21,24)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(1,2)(3,4)(5,6)(7,8)(9,13,15,22)(10,17,12,20)(11,18,14,21)(16,23,19,24)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(1,4)(2,3)(5,7)(6,8)(9,14,15,11)(10,16,12,19)(13,21,22,18)(17,23,20,24)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8672795);