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 | |
4434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,11)(15,17)(16,18)(22,23)(25,26)(27,36)(28,39)(29,30)(31,33)(32,38)(34,37)(35,40);(1,3,5,8)(2,6,7,4)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,31,38)(26,32,33,36)(28,35,37,29)(30,34,40,39);(1,2)(3,4)(5,7)(6,8)(13,21)(16,18)(19,24)(22,23)(26,33)(30,40)(32,36)(34,39);(1,4)(2,3)(5,6)(7,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28)(26,34)(27,35)(29,38)(30,36)(31,37)(32,40)(33,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29,31,35)(26,30,33,40)(27,37,38,28)(32,39,36,34);(1,4)(2,3)(5,6)(7,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30)(26,35)(27,34)(28,36)(29,33)(31,40)(32,37)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8807247);