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 | |
4399 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)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,3)(2,4)(5,7)(6,8)(9,11,15,20)(10,16,17,24)(12,21,23,14)(13,19,18,22)(27,33)(30,37)(34,39)(38,40);(10,17)(11,20)(12,23)(19,22)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(9,12)(10,18)(11,21)(13,17)(14,20)(15,23)(16,22)(19,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,18)(10,12,17,23)(11,22,20,19)(14,24,21,16)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(1,4,5,6)(2,7,8,3)(9,14,15,21)(10,19,17,22)(11,12,20,23)(13,16,18,24)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)Polycyclic presentation:
2^9|b^2=i,c^2=hi,e^2=i,f^2=gi,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, 9637545);