Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,15,16,13)(18,22)(20,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,39)(38,40);(9,10,12,14)(11,13,16,15)(18,22)(19,23)(26,27)(28,35)(29,37)(39,40);(9,11)(10,13)(12,16)(14,15)(17,18,21,22)(19,20,23,24)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,39,38,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,23)(18,20,22,24)(25,28,30,35)(26,31,27,33)(29,38,37,36)(32,40,34,39);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,23)(19,22)(21,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3140849);