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 | |
4564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(2,6)(3,7)(10,14)(11,18)(16,20)(19,24)(26,28)(31,33)(32,36)(38,39);(1,3)(2,4)(5,7)(6,8)(10,14)(15,17)(16,20)(22,23)(27,34)(31,33)(35,40)(38,39);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021624);