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 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(3,8)(9,10,13,12)(11,15,16,14)(25,26)(27,32)(28,35)(29,31)(30,37)(33,38)(34,40)(36,39);(1,2)(3,4)(5,6)(7,8)(18,21)(22,24)(25,27,31,28)(26,32,29,35)(30,34,38,36)(33,39,37,40);(2,6)(4,7)(9,10,13,12)(11,14,16,15)(17,18,20,21)(19,22,23,24)(25,28,31,27)(26,32,29,35)(30,36,38,34)(33,39,37,40);(1,3,5,8)(2,7,6,4)(9,11,13,16)(10,14,12,15)(25,29,31,26)(27,32,28,35)(30,37,38,33)(34,39,36,40);(1,4,5,7)(2,3,6,8)(9,12,13,10)(11,15,16,14)(17,19,20,23)(18,22,21,24)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=hi,d^2=i,e^2=fghi,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, 3753562);