Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,36)(28,40)(29,34)(30,46)(31,35)(32,38)(33,43)(37,50)(39,48)(41,47)(42,51)(44,55)(45,49)(52,53)(54,56);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,29,45,31,39,30,38)(26,32,46,48,35,49,34,36)(28,41,42,53,44,37,43,56)(33,50,55,52,51,47,40,54);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,22)(18,23)(25,28,30,43,31,44,29,42)(26,33,34,40,35,51,46,55)(27,37,38,53,39,41,45,56)(32,47,36,52,49,50,48,54);(9,12)(10,16)(11,18)(13,19)(14,20)(15,22)(17,23)(21,24)(25,29,31,30)(26,34,35,46)(27,38,39,45)(28,42,44,43)(32,48,49,36)(33,40,51,55)(37,53,41,56)(47,54,50,52);(12,19)(16,22)(18,23)(20,24)(27,39)(32,49)(36,48)(37,41)(38,45)(47,50)(52,54)(53,56)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6475938);