Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1237 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)(4,6)(5,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,31,36)(27,37,41,48)(28,42,46,51)(29,34,30,35)(32,40,49,39)(33,45,52,44)(38,47,43,50)(53,55,56,54);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,39,31,41,30,40)(26,32,35,37,36,49,34,48)(28,43,45,54,46,38,44,55)(33,50,51,56,52,47,42,53);(1,3,5,4)(2,7,8,6)(11,17)(14,20)(18,23)(21,24)(25,28,29,45,31,46,30,44)(26,33,35,51,36,52,34,42)(27,38,39,55,41,43,40,54)(32,47,37,53,49,50,48,56);(1,4,5,3)(2,6,8,7)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24);(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,36,35)(27,39,41,40)(28,44,46,45)(32,48,49,37)(33,51,52,42)(38,54,43,55)(47,53,50,56)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6474253);