Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| 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 | |
1298 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,36,31,34,29,35)(27,37,41,32,39,48,40,47)(28,42,44,49,45,50,43,33)(38,52,55,56,53,46,54,51);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,39)(26,32,34,47)(28,38,45,53)(29,41,30,40)(33,46,49,52)(35,48,36,37)(42,56,50,51)(43,55,44,54);(11,17)(14,21)(18,19)(22,23)(25,28,31,45)(26,33,34,49)(27,38,39,53)(29,44,30,43)(32,46,47,52)(35,42,36,50)(37,51,48,56)(40,55,41,54);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(26,34)(32,47)(33,49)(35,36)(37,48)(42,50)(46,52)(51,56);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(26,34)(27,39)(33,49)(35,36)(38,53)(40,41)(42,50)(54,55)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393506);