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 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,15,17)(11,13,20,16)(12,21,24,18)(14,19,22,23);(1,2)(3,8)(4,6)(5,7)(9,11,15,20)(10,16,17,13)(12,22,24,14)(18,23,21,19)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,39)(37,40);(1,3,4,7)(2,5,6,8)(10,17)(13,16)(18,21)(19,23)(26,31)(28,38)(32,40)(34,35);(2,6)(5,8)(9,12,15,24)(10,18,17,21)(11,14,20,22)(13,23,16,19)(25,27)(26,32)(28,35)(29,33)(30,37)(31,40)(34,38)(36,39);(1,3)(2,5)(4,7)(6,8)(9,13,15,16)(10,11,17,20)(12,23,24,19)(14,21,22,18)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,19)(11,12)(13,21)(15,22)(16,18)(17,23)(20,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^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, 8490612);