Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1499 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,28,24,29,22,27)(19,30,21,38,32,44,39,25)(20,33,36,42,37,26,35,43)(31,45,34,40,46,41,48,47);(2,6)(7,8)(17,19,24,32)(18,25,29,38)(20,34,37,48)(21,23,39,22)(26,41,33,45)(27,30,28,44)(31,35,46,36)(40,42,47,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,36,24,37,22,35)(18,26,28,43,29,33,27,42)(19,31,21,34,32,46,39,48)(25,40,30,41,38,47,44,45);(1,4,5,3)(2,8,6,7)(17,21,23,32,24,39,22,19)(18,25,28,30,29,38,27,44)(20,34,36,46,37,48,35,31)(26,40,43,41,33,47,42,45);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,27,29,28)(19,21,32,39)(20,35,37,36)(25,30,38,44)(26,42,33,43)(31,34,46,48)(40,41,47,45)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366511);