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 | |
4384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,17)(13,23)(14,15)(18,24)(19,22)(20,21)(25,26,30,32)(27,31,34,28)(29,37,39,33)(35,36,40,38);(1,3)(2,6)(4,7)(5,8)(10,15)(12,21)(16,22)(18,23)(25,27)(26,31)(28,32)(29,35)(30,34)(33,38)(36,37)(39,40);(10,15)(11,19)(17,20)(18,23)(26,32)(27,34)(33,37)(35,40);(9,11)(10,16)(12,18)(13,20)(14,19)(15,22)(17,24)(21,23)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,38)(39,40);(1,4,5,2)(3,7,8,6)(9,12,14,21)(10,17,15,20)(11,18,19,23)(13,16,24,22)(25,28)(26,27)(29,38)(30,31)(32,34)(33,40)(35,37)(36,39);(1,3)(2,7)(4,6)(5,8)(9,13,14,24)(10,18,15,23)(11,20,19,17)(12,22,21,16)(25,29,30,39)(26,33,32,37)(27,35,34,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=gh,e^2=g,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, 9622648);