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 | |
4341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(12,20)(15,22)(18,24)(21,23)(26,29)(31,33)(32,37)(38,39);(1,3,5,8)(2,6,7,4)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(28,36)(32,37)(34,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,4)(2,3)(5,6)(7,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,29,30,26)(27,33,35,31)(28,37,36,32)(34,39,40,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,f^2=h,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, 9619985);