Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10)(11,15)(12,14)(13,16)(18,20)(22,24)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(1,2,5,4)(3,6,7,8)(9,11,14,13)(10,15,12,16)(19,23)(22,24)(26,30)(27,33)(32,36)(35,37);(9,10,14,12)(11,16,13,15)(17,18,21,20)(19,22,23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,38)(37,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,15,13,16)(17,19,21,23)(18,22,20,24)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(17,20,21,18)(19,24,23,22)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,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, 2705473);