Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,7,4,6)(10,14)(11,17)(16,21)(19,23)(25,26,29,34,30,35,31,36)(27,37,44,47,40,32,39,49)(28,41,45,33,43,50,42,48)(38,52,53,51,54,56,55,46);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,30,40)(26,32,35,37)(28,38,43,54)(29,44,31,39)(33,46,48,51)(34,49,36,47)(41,56,50,52)(42,55,45,53);(1,3,5,4)(2,7,8,6)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28)(26,33)(27,38)(29,42)(30,43)(31,45)(32,46)(34,41)(35,48)(36,50)(37,51)(39,53)(40,54)(44,55)(47,52)(49,56);(1,4,5,3)(2,6,8,7)(12,20)(16,21)(19,23)(22,24);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,31)(26,34,35,36)(27,39,40,44)(28,42,43,45)(32,47,37,49)(33,41,48,50)(38,53,54,55)(46,52,51,56)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ei,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6462104);