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:
(2,5)(6,8)(9,10)(11,12)(13,21)(14,15)(16,22)(17,19)(18,24)(20,23)(25,26)(27,30)(28,36)(29,31)(32,34)(33,38)(35,40)(37,39);(1,2,4,5)(3,6,7,8)(10,15)(12,19)(16,21)(20,24)(25,27,29,34)(26,30,31,32)(28,37,38,35)(33,40,36,39);(10,15)(11,17)(16,21)(18,23)(26,31)(27,34)(33,36)(35,37);(9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,24)(22,23)(25,27)(26,32)(28,35)(29,34)(30,31)(33,40)(36,39)(37,38);(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,23)(22,24)(25,26)(27,32)(28,33)(29,31)(30,34)(35,40)(36,38)(37,39);(1,3,4,7)(2,6,5,8)(9,13,14,22)(10,16,15,21)(11,18,17,23)(12,20,19,24)(25,28)(26,33)(27,35)(29,38)(30,39)(31,36)(32,40)(34,37)Polycyclic presentation:
2^9|b^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7897048);