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 | |
1280 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)(3,7)(4,6)(5,8)(10,14)(15,17)(16,21)(22,23)(25,26,31,43,30,34,29,35)(27,32,44,54,37,47,38,46)(28,39,42,33,41,49,40,48)(36,50,53,45,51,56,52,55);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,30,37)(26,32,34,47)(28,36,41,51)(29,38,31,44)(33,45,48,55)(35,46,43,54)(39,50,49,56)(40,52,42,53);(1,3,5,4)(2,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,41)(26,33,34,48)(27,36,37,51)(29,42,31,40)(32,45,47,55)(35,49,43,39)(38,53,44,52)(46,56,54,50);(1,4,5,3)(2,6,8,7)(12,20)(16,21)(18,24)(22,23);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(26,34)(27,37)(33,48)(35,43)(36,51)(38,44)(39,49)(52,53)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=fi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6459425);