Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907977 |
| 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 | |
1321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,18,21,15)(12,16,22,13)(19,23,24,20)(25,26,30,34)(27,31,36,44)(28,38,39,53)(29,41,42,33)(32,46,47,40)(35,48,49,56)(37,52,51,45)(43,54,55,50);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,18)(12,19,22,24)(13,23,16,20)(25,27,30,36)(26,31,34,44)(28,35,39,49)(29,37,42,51)(32,43,47,55)(33,45,41,52)(38,48,53,56)(40,50,46,54);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,17)(14,22)(15,20)(18,23)(21,24)(25,28,29,40,30,39,42,46)(26,32,33,38,34,47,41,53)(27,35,37,50,36,49,51,54)(31,43,45,48,44,55,52,56);(3,8)(5,7)(10,17)(12,22)(15,18)(19,24)(27,36)(31,44)(35,49)(37,51)(43,55)(45,52)(48,56)(50,54);(2,6)(5,7)(9,13,14,16)(10,12,17,22)(11,20,21,23)(15,24,18,19)(27,36)(28,39)(31,44)(32,47)(37,51)(38,53)(40,46)(45,52)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820989);