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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,31)(27,35,36,49)(28,33,40,47)(29,44,45,34)(32,38,46,39)(37,53,41,56)(42,52,43,51)(48,55,50,54);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(27,36)(28,40)(34,44)(38,39)(42,43)(48,50)(53,56);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,39,30,36,45,38)(26,32,34,35,31,46,44,49)(28,41,43,55,40,37,42,54)(33,50,52,56,47,48,51,53);(1,3,5,4)(2,8,6,7)(12,20)(15,24)(17,19)(21,23)(25,28,30,40)(26,33,31,47)(27,37,36,41)(29,43,45,42)(32,48,46,50)(34,52,44,51)(35,53,49,56)(38,55,39,54);(1,4,5,3)(2,8,6,7)(10,16)(11,18)(15,24)(17,19)(25,29,30,45)(26,34,31,44)(27,38,36,39)(28,42,40,43)(32,49,46,35)(33,51,47,52)(37,54,41,55)(48,53,50,56)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fgi,d^2=f,e^2=f,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, 6960291);