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 | |
4242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,16)(12,13,21,24)(19,23,22,20)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,2)(3,4)(5,6)(7,8)(10,15)(12,21)(16,17)(19,22)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(2,6)(3,7)(10,15)(11,18)(12,21)(20,23)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(3,7)(4,8)(9,11)(10,16)(12,22)(13,23)(14,18)(15,17)(19,21)(20,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,3)(2,4)(5,7)(6,8)(9,12)(10,13)(11,19)(14,21)(15,24)(16,20)(17,23)(18,22)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37);(1,4)(2,3)(5,8)(6,7)(9,13)(10,12)(11,20)(14,24)(15,21)(16,19)(17,22)(18,23)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532628);