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 | |
4570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,18)(16,21)(17,19)(20,23)(22,24)(25,26)(27,32)(28,34)(29,35)(30,38)(31,33)(36,40)(37,39);(1,2,5,6)(3,7,8,4)(10,11)(15,17)(16,18)(22,23)(25,27)(26,32)(28,31)(29,36)(30,37)(33,34)(35,40)(38,39);(1,2)(3,7)(4,8)(5,6)(10,11)(12,19)(13,21)(22,23)(26,33)(27,28)(29,39)(30,40);(2,6)(3,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,27)(26,34,33,32)(29,36,39,38)(30,35,40,37);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,39)(26,35,33,37)(27,36,28,38)(30,32,40,34);(1,4)(2,3)(5,7)(6,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30)(26,36)(27,37)(28,35)(29,32)(31,40)(33,38)(34,39)Polycyclic presentation:
2^9|b^2=ghi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8968616);