Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1358 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)(27,37)(28,42)(29,34)(30,35)(31,36)(32,39)(33,43)(38,46)(40,48)(41,47)(44,50)(45,49)(51,53)(52,55)(54,56);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,29,40,31,41,30,39)(26,32,35,47,36,48,34,37)(28,38,43,53,45,54,44,52)(33,46,42,55,50,56,49,51);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,22)(18,23)(25,28,29,43,31,45,30,44)(26,33,35,42,36,50,34,49)(27,38,40,53,41,54,39,52)(32,46,47,55,48,56,37,51);(9,12)(10,16)(11,18)(13,19)(14,20)(15,22)(17,23)(21,24)(25,29,31,30)(26,34,36,35)(27,39,41,40)(28,43,45,44)(32,47,48,37)(33,49,50,42)(38,52,54,53)(46,55,56,51);(12,19)(16,22)(18,23)(20,24)(25,29,31,30)(26,34,36,35)(27,40,41,39)(28,43,45,44)(32,37,48,47)(33,49,50,42)(38,53,54,52)(46,51,56,55)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fi,d^2=i,e^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^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6475958);