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 | |
4586 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,26)(27,32)(28,30)(29,34)(31,33)(35,39)(36,40)(37,38);(1,3)(2,4)(5,7)(6,8)(9,11,14,19)(10,15,13,17)(12,18,20,21)(16,23,22,24)(25,27,31,36)(26,32,33,40)(28,38,35,29)(30,37,39,34);(1,3)(2,6)(4,8)(5,7)(10,13)(12,20)(15,17)(18,21)(26,33)(27,36)(29,38)(30,39);(1,4,7,6)(2,3,8,5)(9,11)(10,15)(12,21)(13,17)(14,19)(16,24)(18,20)(22,23)(25,28,31,35)(26,30,33,39)(27,29,36,38)(32,34,40,37);(1,5,7,3)(2,4,8,6)(9,12,14,20)(10,16,13,22)(11,18,19,21)(15,23,17,24)(25,29)(26,34)(27,28)(30,32)(31,38)(33,37)(35,36)(39,40);(1,6,7,4)(2,3,8,5)(9,13,14,10)(11,17,19,15)(12,22,20,16)(18,24,21,23)(25,30)(26,35)(27,37)(28,33)(29,32)(31,39)(34,36)(38,40)Polycyclic presentation:
2^9|b^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947712);