Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,34)(27,35,39,49)(28,32,40,46)(29,42,36,33)(31,43,45,38)(37,48,51,56)(41,53,50,47)(44,54,55,52);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,36,43,30,39,29,38)(26,31,42,49,34,45,33,35)(28,37,50,54,40,51,41,52)(32,44,53,56,46,55,47,48);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,28,30,40)(26,32,34,46)(27,37,39,51)(29,41,36,50)(31,44,45,55)(33,47,42,53)(35,48,49,56)(38,52,43,54);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(28,40)(32,46)(37,51)(41,50)(44,55)(47,53)(48,56)(52,54)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6960621);