Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,37,22,27,21,26)(19,28,31,40,30,24,32,39)(20,33,34,25,35,41,36,42)(29,44,47,43,45,48,46,38);(1,2)(3,6)(4,8)(5,7)(10,13)(11,15)(17,19,22,30)(18,24,27,28)(20,29,35,45)(21,32,23,31)(25,38,42,43)(26,40,37,39)(33,48,41,44)(34,47,36,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,20,22,35)(18,25,27,42)(19,29,30,45)(21,34,23,36)(24,38,28,43)(26,41,37,33)(31,46,32,47)(39,48,40,44);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(19,30)(24,28)(29,45)(31,32)(38,43)(39,40)(44,48)(46,47);(1,4,5,3)(2,6,7,8)(17,21,22,23)(18,26,27,37)(19,31,30,32)(20,34,35,36)(24,39,28,40)(25,41,42,33)(29,46,45,47)(38,48,43,44)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=f,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6782174);