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 | |
4538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(11,17)(15,18)(19,23)(22,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,3,5,8)(2,6,7,4)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27)(26,31)(28,30)(29,35)(32,39)(33,34)(36,38)(37,40);(1,3)(2,4)(5,8)(6,7)(10,14)(15,18)(16,20)(22,24)(25,26,30,33)(27,31,28,34)(29,32,38,37)(35,39,36,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9930537);