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 | |
4206 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)(6,8)(9,10,14,17)(11,13,19,15)(12,16,20,23)(18,21,24,22)(25,26)(27,37)(28,33)(29,35)(30,40)(31,36)(32,38)(34,39);(1,2,4,5)(3,6,7,8)(12,20)(16,23)(18,24)(21,22)(25,27)(26,32)(28,30)(29,33)(31,38)(34,40)(35,39)(36,37);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,15,17,13)(12,18,20,24)(16,22,23,21)(25,28,31,39)(26,33,36,34)(27,30,38,35)(29,37,40,32);(2,5)(6,8)(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,24)(22,23)(25,28)(26,34)(27,35)(29,32)(30,38)(31,39)(33,36)(37,40);(1,2)(3,8)(4,5)(6,7)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29,31,40)(26,35,36,30)(27,34,38,33)(28,37,39,32);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,21)(14,15)(16,18)(17,19)(20,22)(23,24)(25,30,31,35)(26,29,36,40)(27,28,38,39)(32,33,37,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10219820);