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 | |
4683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,16,17,12)(13,18,22,23)(19,24,21,20)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2,4,5)(3,6,7,8)(9,11)(10,12)(13,21)(14,17)(15,16)(18,24)(19,22)(20,23)(25,27,31,38)(26,32,36,37)(28,30,34,39)(29,35,33,40);(10,15)(11,17)(13,22)(20,24)(27,38)(30,39)(32,37)(35,40);(9,11)(10,16)(12,15)(13,19)(14,17)(18,24)(20,23)(21,22)(25,28)(26,33)(27,30)(29,36)(31,34)(32,40)(35,37)(38,39);(1,3,4,7)(2,6,5,8)(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,29,31,33)(26,34,36,28)(27,35,38,40)(30,32,39,37);(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,24)(17,21)(25,30)(26,35)(27,28)(29,32)(31,39)(33,37)(34,38)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7719259);