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 | |
4726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26)(27,29)(28,33)(30,40)(31,34)(32,38)(35,37)(36,39);(2,6)(4,7)(10,14)(15,17)(16,20)(22,23)(25,27,31,30)(26,29,34,40)(28,36,38,35)(32,37,33,39);(1,2)(3,4)(5,6)(7,8)(10,14)(11,18)(12,21)(22,23)(25,28,31,38)(26,32,34,33)(27,35,30,36)(29,39,40,37);(1,3)(2,4)(5,8)(6,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,28)(26,33)(27,36)(29,39)(30,35)(31,38)(32,34)(37,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,37)(30,34)(31,40)(32,36)(33,35)(38,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30,31,27)(26,29,34,40)(28,35,38,36)(32,37,33,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=hi,f^2=gh,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, 8894471);