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 | |
1299 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)(25,26,32,47,31,36,30,35)(27,37,29,33,40,49,39,50)(28,41,43,53,45,34,44,52)(38,55,46,51,56,54,42,48);(1,3,4,8)(2,5,6,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,31,40)(26,33,36,50)(28,38,45,56)(29,32,39,30)(34,48,41,51)(35,49,47,37)(42,44,46,43)(52,54,53,55);(2,6)(5,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28)(26,34)(27,38)(29,46)(30,44)(31,45)(32,43)(33,48)(35,53)(36,41)(37,54)(39,42)(40,56)(47,52)(49,55)(50,51);(3,8)(5,7)(11,18)(14,17)(19,24)(22,23)(25,29,30,27,31,39,32,40)(26,33,35,37,36,50,47,49)(28,42,44,56,45,46,43,38)(34,51,53,55,41,48,52,54);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(25,30,31,32)(26,35,36,47)(27,29,40,39)(28,43,45,44)(33,49,50,37)(34,52,41,53)(38,42,56,46)(48,54,51,55)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,d^2=fgi,e^2=hi,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, 7096318);