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 | |
4766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(11,17)(16,23)(18,21)(19,24)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40);(3,7)(5,8)(9,10,14,15)(11,16,17,23)(12,20,22,13)(18,19,21,24)(25,27)(26,32)(28,36)(29,33)(30,35)(31,37)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(10,15)(13,20)(16,23)(19,24)(25,28,31,39)(26,33,35,40)(27,36,37,34)(29,30,38,32);(9,11)(10,16)(12,21)(13,19)(14,17)(15,23)(18,22)(20,24)(25,29)(26,34)(27,33)(28,30)(31,38)(32,39)(35,36)(37,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,13)(11,18)(14,22)(15,20)(16,24)(17,21)(19,23)(25,30)(26,27)(28,38)(29,39)(31,32)(33,36)(34,40)(35,37);(1,2)(3,5)(4,6)(7,8)(9,13)(10,12)(11,19)(14,20)(15,22)(16,21)(17,24)(18,23)(25,27)(26,30)(28,34)(29,33)(31,37)(32,35)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^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, 8833407);