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 | |
4310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,13,19,15)(12,16,20,23)(18,21,24,22)(25,26)(27,29)(28,36)(30,38)(31,32)(33,37)(34,39)(35,40);(1,2,5,6)(3,4,7,8)(12,20)(16,23)(18,24)(21,22)(26,32)(29,33)(30,40)(34,36);(2,6)(3,7)(9,11,14,19)(10,15,17,13)(12,18,20,24)(16,22,23,21)(25,27,31,37)(26,33,32,29)(28,38,39,35)(30,36,40,34);(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,24)(22,23)(25,28,31,39)(26,34,32,36)(27,35,37,38)(29,30,33,40);(1,3)(2,4)(5,7)(6,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29,31,33)(26,27,32,37)(28,30,39,40)(34,35,36,38);(1,4)(2,3)(5,8)(6,7)(9,13)(10,11)(12,21)(14,15)(16,18)(17,19)(20,22)(23,24)(25,30)(26,35)(27,36)(28,33)(29,39)(31,40)(32,38)(34,37)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9148403);