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 | |
4684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,17)(11,19,21,16)(12,20,23,18)(13,22,24,14);(1,2,5,4)(3,7,8,6)(9,11,15,21)(10,16,17,19)(12,22,23,14)(13,18,24,20)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,39)(37,40);(1,3,5,8)(2,6,4,7)(10,17)(13,24)(16,19)(18,20)(26,31)(28,38)(32,40)(34,35);(1,4,5,2)(3,7,8,6)(9,12,15,23)(10,18,17,20)(11,14,21,22)(13,19,24,16)(25,27)(26,32)(28,35)(29,33)(30,37)(31,40)(34,38)(36,39);(9,13)(10,14)(11,20)(12,19)(15,24)(16,23)(17,22)(18,21)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(9,14)(10,13)(11,12)(15,22)(16,18)(17,24)(19,20)(21,23)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8490659);