Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26,30,34)(27,36,38,32)(28,41,44,51)(29,35,31,45)(33,46,50,43)(37,53,42,56)(39,52,40,48)(47,54,49,55);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,32)(28,42)(29,39)(30,38)(31,40)(33,49)(34,36)(35,48)(37,44)(41,53)(43,55)(45,52)(46,54)(47,50)(51,56);(3,8)(5,7)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,28,30,44)(26,33,34,50)(27,37,38,42)(29,43,31,46)(32,47,36,49)(35,51,45,41)(39,54,40,55)(48,53,52,56);(2,6)(5,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(26,34)(27,38)(33,50)(35,45)(37,42)(39,40)(41,51)(54,55);(11,19)(15,17)(18,24)(22,23)(25,29,30,31)(26,35,34,45)(27,39,38,40)(28,43,44,46)(32,48,36,52)(33,51,50,41)(37,54,42,55)(47,53,49,56)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fhi,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6861522);