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 | |
4383 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)(11,16)(12,18)(13,14)(15,17)(19,23)(20,24)(21,22)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(1,2,5,7)(3,4,8,6)(9,11)(10,16)(12,19)(13,15)(14,17)(18,23)(20,21)(22,24)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(1,3,5,8)(2,6,7,4)(10,17)(11,13)(12,20)(22,23)(26,28)(29,38)(31,33)(34,37);(9,12,15,20)(10,18,17,24)(11,19,13,21)(14,22,16,23)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,13,15,11)(10,14,17,16)(12,21,20,19)(18,22,24,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37);(1,4)(2,3)(5,6)(7,8)(9,14)(10,11)(12,22)(13,17)(15,16)(18,19)(20,23)(21,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38)Polycyclic presentation:
2^9|b^2=hi,c^2=ghi,d^2=h,e^2=gh,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, 9157518);