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 | |
1246 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)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39)(41,42,46,49)(43,44,50,47)(45,53,54,48)(51,55,56,52);(1,2,4,6)(3,5,7,8)(10,14)(15,20)(16,21)(22,24)(26,30)(27,34)(28,37)(38,39)(43,50)(44,47)(51,56)(52,55);(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40)(41,43)(42,47)(44,49)(45,51)(46,50)(48,55)(52,53)(54,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39)(41,44,46,47)(42,43,49,50)(45,52,54,55)(48,51,53,56);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)(41,45)(42,48)(43,51)(44,52)(46,54)(47,55)(49,53)(50,56)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=h,e^2=f,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, 4384844);