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 | |
4452 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)(9,10)(11,16)(12,15)(13,17)(14,18)(19,22)(20,23)(21,24)(26,31)(27,34)(29,37)(36,39);(1,2,6,5)(3,4,7,8)(9,11,15,18)(10,16,12,14)(13,21,23,19)(17,24,20,22)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(1,3)(2,4)(5,8)(6,7)(10,12)(11,18)(17,20)(19,21)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,4)(2,7)(3,5)(6,8)(9,12,15,10)(11,14,18,16)(13,20,23,17)(19,24,21,22)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,13,15,23)(10,17,12,20)(11,19,18,21)(14,24,16,22)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(1,5,6,2)(3,8,7,4)(9,14)(10,11)(12,18)(13,22)(15,16)(17,21)(19,20)(23,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=gh,e^2=gh,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, 9633826);