Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1238 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)(9,10,13,15)(11,14,17,21)(12,16,19,22)(18,20,23,24)(25,26,30,34)(27,35,39,51)(28,40,42,32)(29,44,36,33)(31,45,47,38)(37,53,41,50)(43,49,52,55)(46,54,48,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,36,45,30,39,29,38)(26,31,44,51,34,47,33,35)(28,41,52,54,42,37,43,56)(32,48,55,50,40,46,49,53);(2,5)(6,8)(10,15)(11,17)(16,22)(18,23)(25,28,30,42)(26,32,34,40)(27,37,39,41)(29,43,36,52)(31,46,47,48)(33,49,44,55)(35,50,51,53)(38,54,45,56);(1,3,4,7)(2,6,5,8)(12,19)(16,22)(18,23)(20,24);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(28,42)(32,40)(37,41)(43,52)(46,48)(49,55)(50,53)(54,56)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=gh,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6454874);