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 | |
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)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40)(41,42,46,44)(43,49,51,47)(45,48,53,52)(50,55,56,54);(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)(47,49)(48,52)(54,55);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40)(41,43,46,51)(42,47,44,49)(45,50,53,56)(48,54,52,55);(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)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40)(41,44,46,42)(43,49,51,47)(45,48,53,52)(50,54,56,55);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)(41,45)(42,48)(43,50)(44,52)(46,53)(47,54)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=fh,e^2=fg,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, 4778976);