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 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,5)(3,8,4,7)(10,14)(15,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(36,43)(39,46)(41,47)(44,48);(18,20)(23,25)(24,28)(31,32)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(1,3,6,4)(2,7,5,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4,6,3)(2,8,5,7)(9,11,13,12)(10,15,14,16)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,5,6,2)(3,8,4,7)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=i,d^2=gh,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, 4295529);