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 | |
4739 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)(3,7)(4,5)(6,8)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(26,28)(29,37)(31,33)(34,38);(2,5)(6,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(26,28)(31,33)(32,36)(39,40);(1,2)(3,6)(4,5)(7,8)(25,26,30,28)(27,31,35,33)(29,32,37,36)(34,39,38,40);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,27,30,35)(26,31,28,33)(29,38,37,34)(32,40,36,39);(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29,30,37)(26,32,28,36)(27,34,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10330040);