Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,32)(28,39)(29,35)(30,43)(31,34)(33,40)(36,51)(37,47)(38,46)(41,50)(42,48)(44,49)(45,53)(52,55)(54,56);(1,3,5,4)(2,6,8,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,31,38)(26,32,34,46)(28,36,41,54)(29,42,30,37)(33,45,49,55)(35,48,43,47)(39,51,50,56)(40,53,44,52);(1,3,5,4)(2,7,8,6)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,28)(26,33)(27,36)(29,40)(30,44)(31,41)(32,45)(34,49)(35,50)(37,52)(38,54)(39,43)(42,53)(46,55)(47,51)(48,56);(1,4,5,3)(2,6,8,7)(13,21)(16,20)(18,24)(22,23)(26,34)(32,46)(33,49)(35,43)(39,50)(45,55)(47,48)(51,56);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,31,30)(26,35,34,43)(27,37,38,42)(28,40,41,44)(32,47,46,48)(33,50,49,39)(36,52,54,53)(45,51,55,56)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=f,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6474958);