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 | |
4563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(9,10,14,15)(11,16,19,23)(12,17,22,21)(13,20,24,18)(25,26,30,33)(27,31,35,29)(28,32,38,40)(34,37,36,39);(1,3)(2,4)(5,6)(7,8)(10,15)(11,19)(17,21)(18,20)(25,27,30,35)(26,29,33,31)(28,36,38,34)(32,37,40,39);(1,4)(2,6)(3,8)(5,7)(10,15)(11,19)(17,21)(18,20);(1,2,5,8)(3,7,6,4)(9,11)(10,16)(12,20)(13,21)(14,19)(15,23)(17,24)(18,22)(25,27)(26,31)(28,34)(29,33)(30,35)(32,37)(36,38)(39,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,17)(11,20)(13,23)(14,22)(15,21)(16,24)(18,19)(25,28)(26,32)(27,34)(29,39)(30,38)(31,37)(33,40)(35,36);(1,3)(2,7)(4,8)(5,6)(9,13,14,24)(10,18,15,20)(11,21,19,17)(12,16,22,23)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,36,40,34)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854171);