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 | |
4530 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)(9,10,15,17)(11,16,19,24)(12,20,21,18)(13,14,22,23)(25,26)(27,30)(28,37)(29,33)(31,34)(32,36)(35,40)(38,39);(2,5)(3,8)(9,11)(10,16)(12,13)(14,20)(15,19)(17,24)(18,23)(21,22)(25,27)(26,30)(28,29)(31,35)(32,38)(33,37)(34,40)(36,39);(1,3,6,8)(2,7,5,4)(10,17)(11,19)(12,21)(14,23)(27,35)(28,36)(30,40)(32,37);(1,4)(2,3)(5,8)(6,7)(9,12,15,21)(10,18,17,20)(11,13,19,22)(14,16,23,24)(25,28)(26,32)(27,29)(30,38)(31,36)(33,40)(34,37)(35,39);(1,5,6,2)(3,4,8,7)(9,13)(10,14)(11,12)(15,22)(16,20)(17,23)(18,24)(19,21)(25,29)(26,33)(27,36)(28,35)(30,32)(31,39)(34,38)(37,40);(1,4)(2,3)(5,8)(6,7)(9,14)(10,13)(11,20)(12,16)(15,23)(17,22)(18,19)(21,24)(25,30)(26,27)(28,38)(29,32)(31,40)(33,36)(34,35)(37,39)Polycyclic presentation:
2^9|a^2=h,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, 8834445);