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 | |
4647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,16)(12,18)(13,14)(15,20)(17,21)(19,24)(22,23)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,3,5,8)(2,6,7,4)(9,11,15,22)(10,16,20,23)(12,19,21,13)(14,18,24,17)(25,26,29,32)(27,30,35,33)(28,31,37,36)(34,38,40,39);(1,2)(3,4)(5,7)(6,8)(9,12,15,21)(10,17,20,18)(11,13,22,19)(14,16,24,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,18)(11,19)(13,22)(14,23)(15,21)(16,24)(17,20)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,3)(2,4)(5,8)(6,7)(9,13,15,19)(10,14,20,24)(11,21,22,12)(16,17,23,18)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,4)(2,3)(5,6)(7,8)(9,14)(10,19)(11,18)(12,23)(13,20)(15,24)(16,21)(17,22)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gi,d^2=hi,e^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, 8900149);