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 | |
4504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(11,17)(15,18)(19,23)(22,24)(25,26,31,29)(27,33,35,28)(30,37,38,32)(34,36,40,39);(1,3,6,7)(2,4,5,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,31,35)(26,28,29,33)(30,34,38,40)(32,36,37,39);(1,4,6,8)(2,7,5,3)(10,14)(15,18)(16,20)(22,24)(26,29)(28,33)(32,37)(36,39);(1,2,6,5)(3,8,7,4)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,29,31,26)(27,28,35,33)(30,32,38,37)(34,39,40,36);(1,5,6,2)(3,8,7,4)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30,31,38)(26,32,29,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9945654);