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 | |
1303 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(12,20)(15,21)(19,24)(22,23)(25,26,31,32,30,36,29,35)(27,37,44,49,39,33,40,46)(28,34,45,47,43,51,41,52)(38,50,54,56,42,48,53,55);(1,3,4,8)(2,5,6,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(26,32)(28,41)(29,31)(33,46)(35,36)(37,49)(38,53)(40,44)(42,54)(43,45)(47,52)(55,56);(2,6)(5,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,27,30,39)(26,33,36,37)(28,42,43,38)(29,40,31,44)(32,46,35,49)(34,50,51,48)(41,54,45,53)(47,56,52,55);(3,8)(5,7)(11,18)(14,17)(19,24)(22,23)(25,28,29,41,30,43,31,45)(26,34,35,52,36,51,32,47)(27,38,40,53,39,42,44,54)(33,48,49,56,37,50,46,55);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(27,39)(28,43)(33,37)(34,51)(40,44)(41,45)(46,49)(47,52)Polycyclic presentation:
2^9|a^2=fi,b^2=ghi,c^2=i,d^2=fgi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,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, 7096329);