Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1294 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,21)(13,17)(14,16)(15,18)(20,23)(22,24)(25,26,30,35)(27,36,41,48)(28,42,46,51)(29,33,37,34)(31,40,49,39)(32,44,52,45)(38,50,43,47)(53,54,56,55);(2,5)(6,8)(9,11,15,14)(10,16,18,19)(12,22,13,20)(17,24,21,23)(25,27,37,40,30,41,29,39)(26,31,33,48,35,49,34,36)(28,43,44,55,46,38,45,54)(32,50,51,53,52,47,42,56);(1,3,4,7)(2,6,5,8)(9,12,15,13)(10,17,18,21)(11,20,14,22)(16,23,19,24)(25,28,29,45,30,46,37,44)(26,32,34,42,35,52,33,51)(27,38,39,55,41,43,40,54)(31,47,36,53,49,50,48,56);(9,13,15,12)(10,17,18,21)(11,20,14,22)(16,24,19,23)(25,29,30,37)(26,33,35,34)(27,39,41,40)(28,44,46,45)(31,48,49,36)(32,42,52,51)(38,54,43,55)(47,53,50,56);(9,14,15,11)(10,16,18,19)(12,22,13,20)(17,23,21,24)(25,29,30,37)(26,33,35,34)(27,40,41,39)(28,45,46,44)(31,36,49,48)(32,51,52,42)(38,54,43,55)(47,53,50,56)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6464096);