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 | |
4477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,14)(11,15,17,18)(12,20,21,16)(19,24,23,22)(25,26,30,32)(27,33,34,28)(29,31,37,38)(35,39,40,36);(1,2,5,6)(3,7,8,4)(10,14)(15,18)(16,20)(22,24)(25,26)(27,33)(28,34)(29,31)(30,32)(35,39)(36,40)(37,38);(1,2)(3,7)(4,8)(5,6)(10,14)(11,17)(12,21)(22,24)(27,34)(28,33)(29,37)(31,38);(2,6)(3,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,34)(26,28,32,33)(29,35,37,40)(31,36,38,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,29)(26,31)(27,35)(28,36)(30,37)(32,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=gh,f^2=h,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, 8968793);