Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,29,33)(27,35,38,31)(28,32,41,47)(30,34,42,48)(36,49,51,43)(37,46,40,44)(39,45,52,55)(50,56,53,54);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(27,36)(28,39)(30,42)(31,43)(32,45)(34,48)(35,49)(37,40)(38,51)(41,52)(44,46)(47,55);(2,6)(7,8)(25,27,30,36,29,38,42,51)(26,31,34,43,33,35,48,49)(28,40,39,53,41,37,52,50)(32,46,45,56,47,44,55,54);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(28,41)(32,47)(37,40)(39,52)(44,46)(45,55)(50,53)(54,56);(1,4,5,3)(2,8,6,7)(11,19)(14,17)(18,24)(22,23)(25,28,30,39,29,41,42,52)(26,32,34,45,33,47,48,55)(27,37,36,50,38,40,51,53)(31,44,43,54,35,46,49,56)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=h,d^2=fgi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6748804);