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 |
| prime radical | #35730 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4542 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)(11,19)(12,23)(13,18)(14,15)(16,22)(17,21)(20,24)(25,26,31,35)(27,36,38,32)(28,33,39,30)(29,40,37,34);(1,2,5,6)(3,7,8,4)(9,11,15,22)(10,16,14,19)(12,13,21,20)(17,18,23,24)(25,27)(26,32)(28,37)(29,39)(30,34)(31,38)(33,40)(35,36);(1,2)(3,7)(4,8)(5,6)(10,14)(12,21)(13,20)(16,19)(25,28,31,39)(26,33,35,30)(27,37,38,29)(32,40,36,34);(2,6)(3,8)(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,29,31,37)(26,34,35,40)(27,28,38,39)(30,32,33,36);(1,3)(2,4)(5,8)(6,7)(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,28)(26,33)(27,29)(30,35)(31,39)(32,34)(36,40)(37,38);(1,4)(2,3)(5,7)(6,8)(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,30,31,33)(26,28,35,39)(27,34,38,40)(29,32,37,36)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gh,e^2=h,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, 8968662);