Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907977 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,12,13)(11,15,16,14)(17,18,22,26)(19,28,30,24)(20,33,34,40)(21,27,23,37)(25,36,41,35)(29,44,46,48)(31,43,32,39)(38,45,42,47);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,28)(20,29,34,46)(21,31,23,32)(25,38,41,42)(27,39,37,43)(33,48,40,44)(35,45,36,47);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,20,21,36,22,34,23,35)(18,25,37,40,26,41,27,33)(19,29,31,47,30,46,32,45)(24,38,43,44,28,42,39,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(18,26)(19,30)(20,34)(27,37)(31,32)(35,36)(38,42)(44,48);(2,6)(5,7)(17,21,22,23)(18,27,26,37)(19,31,30,32)(20,35,34,36)(24,39,28,43)(25,40,41,33)(29,45,46,47)(38,44,42,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6818266);