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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26,31,35)(27,32,37,46)(28,39,42,53)(29,34,30,44)(33,40,49,41)(36,50,51,56)(38,47,43,48)(45,54,55,52);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,27,31,37)(26,32,35,46)(28,36,42,51)(29,43,30,38)(33,45,49,55)(34,48,44,47)(39,50,53,56)(40,54,41,52);(1,3)(2,5)(4,8)(6,7)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28,30,41,31,42,29,40)(26,33,34,39,35,49,44,53)(27,36,38,52,37,51,43,54)(32,45,48,50,46,55,47,56);(3,8)(5,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(27,37)(32,46)(36,51)(38,43)(45,55)(47,48)(50,56)(52,54);(2,6)(5,7)(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(25,29,31,30)(26,34,35,44)(27,38,37,43)(28,40,42,41)(32,47,46,48)(33,39,49,53)(36,52,51,54)(45,56,55,50)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822300);