Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1242 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)(12,19)(16,21)(17,23)(22,24)(26,30)(28,35)(31,36)(34,39)(41,42,46,43)(44,47,52,49)(45,50,53,48)(51,55,56,54);(1,2,4,6)(3,5,7,8)(10,14)(15,20)(16,21)(22,24)(26,30)(27,33)(32,37)(34,39)(42,43)(45,53)(47,49)(51,56);(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40)(41,43,46,42)(44,47,52,49)(45,50,53,48)(51,54,56,55);(1,3,4,7)(2,5,6,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40)(41,44)(42,47)(43,49)(45,51)(46,52)(48,54)(50,55)(53,56);(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)(41,45,46,53)(42,48,43,50)(44,51,52,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4422501);