Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(11,15)(14,16)(17,18,21,19)(20,23,24,22)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(1,3)(2,4)(5,8)(6,7)(10,13)(14,16)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(2,6)(4,7)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3)(2,7)(4,6)(5,8)(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4,5,7)(2,8,6,3)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=i,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3454248);