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 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,7,5)(9,10)(11,15)(12,18)(13,21)(14,16)(17,20)(19,24)(22,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,20)(10,15,16,17)(12,21,23,19)(13,22,24,18)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(1,3)(2,5)(4,8)(6,7)(10,16)(11,20)(18,22)(19,21);(1,4,6,5)(2,7,8,3)(9,11)(10,17)(12,21)(13,22)(14,20)(15,16)(18,24)(19,23)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,18)(11,21)(13,15)(14,23)(16,22)(17,24)(19,20)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,5,6,4)(2,7,8,3)(9,13,14,24)(10,19,16,21)(11,22,20,18)(12,17,23,15)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=g,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, 9844956);