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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(4,8)(17,18,21,19)(20,23,24,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,2,5,6)(3,8,7,4)(10,13)(14,16)(26,29)(27,33)(32,38)(34,36);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22)(25,27,30,33)(26,31,29,35)(28,36,37,34)(32,40,38,39);(1,2)(3,4)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)(25,28,30,37)(26,32,29,38)(27,34,33,36)(31,39,35,40);(1,4)(2,3)(5,8)(6,7)(25,29,30,26)(27,35,33,31)(28,32,37,38)(34,39,36,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fi,d^2=ghi,e^2=f,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, 3096637);