Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(17,18,22,27)(19,29,32,24)(20,33,36,44)(21,28,23,26)(25,38,42,35)(30,45,34,48)(31,40,37,43)(39,47,41,46);(2,5)(3,7)(9,10,12,13)(11,15,16,14)(17,19,22,32)(18,24,27,29)(20,34,36,30)(21,37,23,31)(25,41,42,39)(26,43,28,40)(33,45,44,48)(35,46,38,47);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,22,36)(18,25,27,42)(19,30,32,34)(21,35,23,38)(24,39,29,41)(26,33,28,44)(31,46,37,47)(40,45,43,48);(2,5)(6,8)(17,21,22,23)(18,26,27,28)(19,31,32,37)(20,35,36,38)(24,40,29,43)(25,33,42,44)(30,46,34,47)(39,45,41,48);(9,11,12,16)(10,14,13,15)(18,27)(24,29)(25,42)(26,28)(33,44)(39,41)(40,43)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=f,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6707763);