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 | |
4407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,21,24,23)(25,26,30,32)(27,31,33,28)(29,34,38,39)(35,37,40,36);(1,2)(3,6)(4,7)(5,8)(25,27,30,33)(26,31,32,28)(29,35,38,40)(34,37,39,36);(1,3,5,4)(2,6,8,7)(10,15)(11,18)(17,20)(19,24)(26,32)(27,33)(34,39)(35,40);(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,40)(38,39);(1,4,5,3)(2,7,8,6)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28,30,31)(26,33,32,27)(29,37,38,36)(34,35,39,40);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,29,30,38)(26,34,32,39)(27,35,33,40)(28,36,31,37)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7848350);