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 |
| prime radical | #45560 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26,32,35)(27,36,33,28)(29,39,40,31)(30,34,38,37);(1,2)(3,6)(4,7)(5,8)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,27,32,33)(26,28,35,36)(29,38,40,30)(31,37,39,34);(1,3)(2,5)(4,8)(6,7)(25,28,32,36)(26,33,35,27)(29,34,40,37)(30,31,38,39);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,29)(26,31)(27,30)(28,34)(32,40)(33,38)(35,39)(36,37);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,30)(26,34)(27,29)(28,31)(32,38)(33,40)(35,37)(36,39);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,31)(26,29)(27,37)(28,38)(30,36)(32,39)(33,34)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003524);