Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,8)(10,14)(15,16)(17,18,21,22)(19,25,27,23)(20,24,29,30)(26,32,28,31)(35,41)(36,43)(38,44)(39,46);(18,22)(23,25)(24,30)(31,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,2,5,6)(3,7,8,4)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,28)(21,27)(22,25)(24,32)(26,29)(30,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3)(2,7)(4,6)(5,8)(9,11,13,12)(10,15,14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,32)(29,30)(33,35,37,41)(34,38,40,44)(36,42,43,47)(39,45,46,48);(1,4,5,7)(2,3,6,8)(9,12,13,11)(10,16,14,15)(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)(33,36)(34,39)(35,42)(37,43)(38,45)(40,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fhi,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4304123);