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 | |
4186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,21)(14,23)(15,16)(17,22)(18,20)(19,24)(25,26)(27,35)(28,33)(29,39)(30,32)(31,36)(34,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,19)(10,13,16,24)(12,20,22,14)(17,23,21,18)(25,27,30,36)(26,31,32,35)(28,37,38,29)(33,39,40,34);(10,16)(11,19)(12,22)(18,23)(26,32)(31,35)(33,40)(34,39);(9,12)(10,17)(11,20)(13,23)(14,19)(15,22)(16,21)(18,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,11)(12,23)(14,21)(15,24)(16,19)(17,20)(18,22)(25,28,30,38)(26,33,32,40)(27,29,36,37)(31,34,35,39);(1,2)(3,5)(4,7)(6,8)(9,14,15,20)(10,18,16,23)(11,12,19,22)(13,17,24,21)(25,29)(26,34)(27,28)(30,37)(31,33)(32,39)(35,40)(36,38)Polycyclic presentation:
2^9|b^2=h,e^2=hi,f^2=i,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, 9623500);