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 | #35729 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,12,21,16)(13,18,23,22)(14,24,20,19)(25,26)(27,38)(28,33)(29,35)(30,36)(31,37)(32,40)(34,39);(1,2,5,6)(3,7,8,4)(9,11,15,21)(10,16,17,12)(13,20,23,14)(18,24,22,19)(25,27)(26,32)(28,29)(30,33)(31,40)(34,35)(36,39)(37,38);(1,2)(3,7)(4,8)(5,6)(10,17)(12,16)(13,23)(14,20)(25,28,31,39)(26,33,37,34)(27,29,40,36)(30,38,35,32);(2,6)(3,8)(9,12,15,16)(10,11,17,21)(13,24,23,19)(14,22,20,18)(25,28)(26,34)(27,36)(29,40)(30,32)(31,39)(33,37)(35,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,18)(11,20)(12,19)(14,21)(15,23)(16,24)(17,22)(25,29,31,36)(26,35,37,30)(27,39,40,28)(32,33,38,34);(1,4)(2,3)(5,7)(6,8)(9,14,15,20)(10,19,17,24)(11,13,21,23)(12,22,16,18)(25,30,31,35)(26,36,37,29)(27,33,40,34)(28,32,39,38)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,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, 8968667);