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 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,15,16,13)(17,18,21,20)(19,22,23,24)(25,26)(27,34)(28,31)(29,32)(30,36)(33,38)(35,39)(37,40);(11,16)(13,15)(19,23)(22,24)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(9,10)(11,13)(12,14)(15,16)(17,19,21,23)(18,22,20,24)(25,28,31,26)(27,32,29,34)(30,33,38,36)(35,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,24,23,22)(25,29,31,27)(26,32,28,34)(30,35,38,37)(33,40,36,39);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,14)(15,16)(25,30,31,38)(26,33,28,36)(27,35,29,37)(32,39,34,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=h,d^2=fgh,e^2=gi,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, 3549940);