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 | |
4394 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)(3,4)(5,8)(6,7)(9,10)(11,17)(12,21)(13,15)(14,18)(16,19)(20,22)(23,24)(25,26,32,29)(27,36,38,33)(28,39,30,34)(31,40,37,35);(1,3,5,7)(2,4,8,6)(9,11,16,13)(10,17,19,15)(12,20,14,23)(18,24,21,22)(25,27)(26,33)(28,37)(29,36)(30,31)(32,38)(34,35)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12,16,14)(10,18,19,21)(11,20,13,23)(15,22,17,24)(25,28,32,30)(26,34,29,39)(27,37,38,31)(33,35,36,40);(1,2,5,8)(3,4,7,6)(9,13,16,11)(10,15,19,17)(12,20,14,23)(18,24,21,22)(25,29,32,26)(27,36,38,33)(28,34,30,39)(31,40,37,35);(1,3)(2,4)(5,7)(6,8)(9,14,16,12)(10,18,19,21)(11,20,13,23)(15,24,17,22)(25,30,32,28)(26,34,29,39)(27,37,38,31)(33,40,36,35);(1,4)(2,7)(3,8)(5,6)(9,15)(10,11)(12,22)(13,19)(14,24)(16,17)(18,20)(21,23)(25,31,32,37)(26,35,29,40)(27,30,38,28)(33,34,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=hi,f^2=g,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, 9633269);