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)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40)(41,42,46,44)(43,49,50,47)(45,52,53,48)(51,54,56,55);(1,2,4,6)(3,5,7,8)(10,14)(15,20)(16,21)(22,24)(27,33)(30,36)(34,39)(37,40)(42,44)(43,50)(45,53)(54,55);(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40)(41,43,46,50)(42,47,44,49)(45,51,53,56)(48,54,52,55);(1,3,4,7)(2,5,6,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40)(41,44,46,42)(43,47,50,49)(45,52,53,48)(51,54,56,55);(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)(41,45,46,53)(42,48,44,52)(43,51,50,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=fghi,e^2=fgh,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, 4422548);