Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | |
1276 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,31,36)(27,32,39,47)(28,40,43,33)(29,34,44,50)(30,35,38,46)(37,51,53,45)(41,54,52,48)(42,55,56,49);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,32)(28,41)(29,38)(30,44)(31,39)(33,48)(34,46)(35,50)(36,47)(37,42)(40,54)(43,52)(45,49)(51,55)(53,56);(1,3,5,4)(2,6,7,8)(25,28,29,42,31,43,44,56)(26,33,34,49,36,40,50,55)(27,37,30,41,39,53,38,52)(32,45,35,48,47,51,46,54);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,31,44)(26,34,36,50)(27,38,39,30)(28,42,43,56)(32,46,47,35)(33,49,40,55)(37,52,53,41)(45,54,51,48);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(22,23)(25,30,44,27,31,38,29,39)(26,35,50,32,36,46,34,47)(28,41,56,37,43,52,42,53)(33,48,55,45,40,54,49,51)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=ghi,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744455);