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 | |
4528 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)(3,4)(5,6)(7,8)(11,18)(16,19)(20,23)(21,24)(25,26,30,28)(27,32,35,31)(29,36,37,33)(34,38,40,39);(1,3,6,7)(2,4,5,8)(9,10,14,15)(11,19,18,16)(12,13,17,22)(20,24,23,21)(25,26)(27,31)(28,30)(29,33)(32,35)(34,38)(36,37)(39,40);(1,4,6,8)(2,7,5,3)(10,15)(12,17)(16,19)(20,23)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(1,2,6,5)(3,8,7,4)(9,11,14,18)(10,16,15,19)(12,20,17,23)(13,21,22,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,13)(11,20)(14,17)(15,22)(16,21)(18,23)(19,24)(25,28,30,26)(27,31,35,32)(29,33,37,36)(34,39,40,38);(1,5,6,2)(3,8,7,4)(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,24)(19,20)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=g,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, 9949757);