Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1278 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)(4,6)(5,8)(10,14)(15,17)(16,21)(22,23)(25,26,30,34,29,33,42,48)(27,35,38,51,37,31,44,46)(28,39,41,47,40,49,43,32)(36,52,54,45,53,56,55,50);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,29,37)(26,31,33,35)(28,36,40,53)(30,38,42,44)(32,45,47,50)(34,46,48,51)(39,56,49,52)(41,54,43,55);(1,3,5,4)(2,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,29,40)(26,32,33,47)(27,36,37,53)(30,43,42,41)(31,45,35,50)(34,49,48,39)(38,55,44,54)(46,52,51,56);(1,4,5,3)(2,6,8,7)(12,20)(16,21)(18,24)(22,23);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(26,33)(27,37)(32,47)(34,48)(36,53)(38,44)(39,49)(54,55)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=ei,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6459399);