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 | |
4800 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)(9,10)(11,14)(12,21)(13,17)(15,18)(16,20)(19,24)(22,23)(25,26)(27,29)(28,30)(31,33)(32,36)(34,38)(35,39)(37,40);(2,6)(4,7)(9,11)(10,14)(12,13)(15,19)(16,22)(17,21)(18,24)(20,23)(25,27,31,36)(26,32,33,29)(28,34,37,39)(30,35,40,38);(1,2,5,6)(3,7,8,4)(11,19)(12,20)(14,24)(16,21)(28,37)(30,40)(34,39)(35,38);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,13)(14,22)(15,20)(17,24)(18,21)(19,23)(25,28)(26,30)(27,34)(29,38)(31,37)(32,35)(33,40)(36,39);(1,4,5,7)(2,3,6,8)(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,22)(21,24)(25,29,31,32)(26,27,33,36)(28,35,37,38)(30,39,40,34);(1,3)(2,7)(4,6)(5,8)(9,14)(10,11)(12,22)(13,16)(15,24)(17,20)(18,19)(21,23)(25,30)(26,28)(27,35)(29,39)(31,40)(32,34)(33,37)(36,38)Polycyclic presentation:
2^9|b^2=gh,c^2=h,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, 8834274);