Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(5,8)(10,14)(11,17)(16,21)(18,23)(27,33)(28,35)(31,36)(32,37)(41,42,47,46)(43,49,44,48)(45,50,53,54)(51,56,52,55);(11,17)(15,20)(18,23)(22,24)(26,30)(31,36)(32,37)(38,40)(41,43,47,44)(42,48,46,49)(45,52,53,51)(50,56,54,55);(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40)(41,44,47,43)(42,49,46,48)(45,52,53,51)(50,56,54,55);(1,3)(2,5)(4,7)(6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40)(41,45,47,53)(42,50,46,54)(43,51,44,52)(48,55,49,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)(41,46,47,42)(43,49,44,48)(45,50,53,54)(51,55,52,56)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fi,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4787371);