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 | |
4476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,17)(11,16,12,22)(13,18,23,21)(14,20,24,19)(27,33)(31,37)(34,39)(38,40);(2,3)(6,7)(9,11)(10,16)(12,15)(13,20)(14,21)(17,22)(18,24)(19,23)(26,30)(27,33)(32,35)(34,39);(1,3,5,2)(4,7,8,6)(10,17)(11,12)(18,21)(19,20)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,4,5,8)(2,6,3,7)(9,12,15,11)(10,16,17,22)(13,19,23,20)(14,18,24,21)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(9,13)(10,18)(11,20)(12,19)(14,22)(15,23)(16,24)(17,21)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(9,14,15,24)(10,19,17,20)(11,21,12,18)(13,16,23,22)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=i,f^2=h,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, 9852802);