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 | |
4421 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)(3,7)(10,15)(11,18)(13,21)(20,23)(25,26,30,32)(27,28,34,36)(29,31,39,38)(33,35,37,40);(1,2,6,5)(3,4,7,8)(9,10,14,15)(11,12,18,16)(13,17,21,22)(19,20,24,23)(25,27)(26,28)(29,37)(30,34)(31,40)(32,36)(33,39)(35,38);(1,3)(2,4)(5,8)(6,7)(9,11,14,18)(10,16,15,12)(13,19,21,24)(17,23,22,20)(25,28,30,36)(26,27,32,34)(29,35,39,40)(31,33,38,37);(1,4)(2,7)(3,5)(6,8)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,24)(22,23)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,38)(39,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,20)(14,16)(15,18)(17,19)(21,23)(22,24)(25,29)(26,31)(27,33)(28,35)(30,39)(32,38)(34,37)(36,40);(1,5,6,2)(3,8,7,4)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,26)(27,28)(29,38)(30,32)(31,39)(33,40)(34,36)(35,37)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9627594);