Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,36)(28,40)(29,34)(30,46)(31,35)(32,38)(33,43)(37,50)(39,48)(41,47)(42,52)(44,49)(45,51)(53,55)(54,56);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,30,38,31,39,29,44)(26,32,34,36,35,49,46,48)(28,41,43,54,42,37,45,55)(33,50,40,56,51,47,52,53);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(12,19)(20,24)(25,28,30,43,31,42,29,45)(26,33,34,40,35,51,46,52)(27,37,38,55,39,41,44,54)(32,47,36,53,49,50,48,56);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(28,42)(33,51)(37,41)(40,52)(43,45)(47,50)(53,56)(54,55);(11,17)(14,21)(18,23)(20,24)(25,29,31,30)(26,34,35,46)(27,38,39,44)(28,43,42,45)(32,48,49,36)(33,52,51,40)(37,54,41,55)(47,53,50,56)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6433706);