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 | |
4534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10,15,18)(11,16,19,24)(12,22,20,14)(13,17,23,21)(25,26,31,33)(27,32,35,40)(28,36,37,34)(29,30,38,39);(2,5)(3,8)(9,11)(10,16)(12,13)(14,21)(15,19)(17,22)(18,24)(20,23)(25,27)(26,32)(28,29)(30,36)(31,35)(33,40)(34,39)(37,38);(1,3,6,8)(2,7,5,4)(11,19)(12,20)(14,22)(16,24)(26,33)(27,35)(28,37)(30,39);(1,4)(2,3)(5,8)(6,7)(9,12)(10,14)(11,13)(15,20)(16,21)(17,24)(18,22)(19,23)(25,28,31,37)(26,34,33,36)(27,29,35,38)(30,32,39,40);(1,5,6,2)(3,4,8,7)(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,21)(22,24)(25,29)(26,30)(27,28)(31,38)(32,36)(33,39)(34,40)(35,37);(1,4)(2,3)(5,8)(6,7)(9,14)(10,12)(11,21)(13,16)(15,22)(17,19)(18,20)(23,24)(25,30)(26,29)(27,36)(28,32)(31,39)(33,38)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835965);