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 | |
4497 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)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26)(27,36)(28,39)(29,30)(31,33)(32,38)(34,37)(35,40);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27,31,38)(26,32,33,36)(28,30,37,40)(29,39,35,34);(2,6)(3,7)(13,21)(16,23)(18,24)(20,22)(26,33)(29,35)(32,36)(34,39);(3,7)(4,8)(9,12,14,15)(10,11,17,19)(13,22,21,20)(16,24,23,18)(25,28)(26,34)(27,30)(29,32)(31,37)(33,39)(35,36)(38,40);(1,3)(2,4)(5,7)(6,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,31,35)(26,30,33,40)(27,34,38,39)(28,36,37,32);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,30)(26,35)(27,37)(28,38)(29,33)(31,40)(32,34)(36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8972079);