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 | |
4499 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)(13,17)(16,18)(19,22)(20,21)(23,24)(25,26,30,34)(27,31,37,32)(28,33,39,29)(35,40,38,36);(1,2,5,6)(3,7,8,4)(9,11,15,18)(10,16,12,14)(13,19,21,24)(17,23,20,22)(25,26)(27,31)(28,29)(30,34)(32,37)(33,39)(35,36)(38,40);(1,2)(3,7)(4,8)(5,6)(10,12)(13,21)(14,16)(19,24)(25,27,30,37)(26,31,34,32)(28,38,39,35)(29,40,33,36);(2,6)(3,8)(9,12,15,10)(11,16,18,14)(13,17,21,20)(19,22,24,23)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,40)(35,39);(1,3)(2,4)(5,8)(6,7)(9,13,15,21)(10,17,12,20)(11,19,18,24)(14,22,16,23)(25,28)(26,33)(27,35)(29,34)(30,39)(31,40)(32,36)(37,38);(1,4)(2,3)(5,7)(6,8)(9,14,15,16)(10,18,12,11)(13,22,21,23)(17,24,20,19)(25,29,30,33)(26,28,34,39)(27,36,37,40)(31,35,32,38)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,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, 8968658);