Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,31,35)(27,36,38,52)(28,41,45,56)(29,34,30,46)(32,39,48,40)(33,43,50,44)(37,49,42,47)(51,53,55,54);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,40,31,38,29,39)(26,32,34,36,35,48,46,52)(28,42,44,53,45,37,43,54)(33,49,41,55,50,47,56,51);(1,3,5,4)(2,7,8,6)(11,17)(14,20)(18,23)(21,24)(25,28,30,44,31,45,29,43)(26,33,34,41,35,50,46,56)(27,37,40,54,38,42,39,53)(32,47,36,51,48,49,52,55);(1,4,5,3)(2,6,8,7)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(32,48)(36,52)(37,42)(39,40)(47,49)(51,55)(53,54);(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,46)(27,39,38,40)(28,43,45,44)(32,36,48,52)(33,41,50,56)(37,53,42,54)(47,51,49,55)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6474155);