Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,34)(27,35,36,48)(28,32,43,52)(29,44,45,33)(31,38,46,39)(37,53,40,56)(41,51,42,50)(47,55,49,54);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,36)(31,46)(35,48)(37,40)(38,39)(47,49)(53,56)(54,55);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,39,30,36,45,38)(26,31,33,35,34,46,44,48)(28,40,42,55,43,37,41,54)(32,49,51,56,52,47,50,53);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,28,30,43)(26,32,34,52)(27,37,36,40)(29,42,45,41)(31,47,46,49)(33,51,44,50)(35,53,48,56)(38,55,39,54);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,30,45)(26,33,34,44)(27,38,36,39)(28,41,43,42)(31,48,46,35)(32,50,52,51)(37,54,40,55)(47,53,49,56)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fg,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6960794);