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 | |
4344 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)(11,17)(16,23)(18,21)(19,24)(25,26,31,33)(27,32,36,38)(28,37,35,30)(29,34,39,40);(2,5)(3,8)(9,10,14,15)(11,16,17,23)(12,20,22,13)(18,19,21,24)(25,27)(26,32)(28,34)(29,30)(31,36)(33,38)(35,40)(37,39);(1,3,6,8)(2,7,5,4)(10,15)(13,20)(16,23)(19,24)(26,33)(27,36)(30,37)(34,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,21)(13,19)(14,17)(15,23)(18,22)(20,24)(25,28,31,35)(26,30,33,37)(27,34,36,40)(29,38,39,32);(1,5,6,2)(3,4,8,7)(9,12)(10,13)(11,18)(14,22)(15,20)(16,24)(17,21)(19,23)(25,29)(26,34)(27,30)(28,32)(31,39)(33,40)(35,38)(36,37);(1,4)(2,3)(5,8)(6,7)(9,13)(10,12)(11,19)(14,20)(15,22)(16,21)(17,24)(18,23)(25,30)(26,35)(27,29)(28,33)(31,37)(32,40)(34,38)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=hi,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,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, 8833516);