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 | |
4631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,26)(27,34)(28,30)(29,33)(31,35)(32,38)(36,37)(39,40);(1,2)(3,7)(4,8)(5,6)(25,27)(26,32)(28,39)(29,36)(30,37)(31,38)(33,40)(34,35);(10,15)(13,22)(16,23)(20,24)(25,28,31,29)(26,30,35,33)(27,36,38,39)(32,40,34,37);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,29,31,28)(26,33,35,30)(27,36,38,39)(32,40,34,37);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,30,31,33)(26,29,35,28)(27,37,38,40)(32,36,34,39);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982513);