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 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,15)(13,21)(16,22)(17,23)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,10,14,15)(11,16,19,22)(12,17,20,23)(13,18,21,24);(10,15)(12,20)(16,22)(18,24)(25,26,29,32)(27,30,35,33)(28,31,37,36)(34,38,40,39);(1,3,5,4)(2,7,6,8)(9,11,14,19)(10,16,15,22)(12,18,20,24)(13,17,21,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,12,14,20)(10,17,15,23)(11,18,19,24)(13,16,21,22)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(9,13,14,21)(10,18,15,24)(11,17,19,23)(12,16,20,22)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550684);