Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1407 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)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,34)(27,36,38,32)(28,41,43,55)(29,35,31,46)(33,44,50,45)(37,52,42,56)(39,51,40,48)(47,54,49,53);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,38)(26,32,34,36)(28,42,43,37)(29,39,31,40)(33,49,50,47)(35,48,46,51)(41,52,55,56)(44,54,45,53);(3,8)(5,7)(12,21)(15,20)(18,24)(22,23)(25,28,31,45,30,43,29,44)(26,33,35,41,34,50,46,55)(27,37,40,54,38,42,39,53)(32,47,48,56,36,49,51,52);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,34)(27,38)(28,43)(35,46)(39,40)(44,45)(47,49)(52,56);(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,34,46)(27,39,38,40)(28,44,43,45)(32,48,36,51)(33,41,50,55)(37,53,42,54)(47,56,49,52)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=g,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855662);