Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1346 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,33)(29,34)(30,44)(31,35)(32,39)(37,50)(38,46)(40,51)(41,47)(42,48)(43,49)(45,53)(52,56)(54,55);(2,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,39,31,40,29,38)(26,32,34,36,35,46,44,51)(28,37,42,53,41,52,43,54)(33,45,49,50,47,55,48,56);(1,3,4,8)(2,6,5,7)(11,17)(12,19)(14,20)(15,22)(28,41)(33,47)(37,52)(42,43)(45,55)(48,49)(50,56)(53,54);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,28,31,41)(26,33,35,47)(27,37,40,52)(29,43,30,42)(32,45,46,55)(34,49,44,48)(36,50,51,56)(38,54,39,53);(11,17)(14,20)(18,23)(21,24)(25,29,31,30)(26,34,35,44)(27,38,40,39)(28,42,41,43)(32,36,46,51)(33,48,47,49)(37,53,52,54)(45,56,55,50)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6435753);