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)(17,18,23,28,22,26,21,27)(19,29,32,41,31,24,36,39)(20,33,34,40,35,42,37,25)(30,44,47,38,45,48,46,43);(1,2)(3,6)(4,8)(5,7)(10,13)(11,15)(17,19,22,31)(18,24,26,29)(20,30,35,45)(21,36,23,32)(25,38,40,43)(27,41,28,39)(33,48,42,44)(34,47,37,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,20,22,35)(18,25,26,40)(19,30,31,45)(21,34,23,37)(24,38,29,43)(27,33,28,42)(32,46,36,47)(39,44,41,48);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(18,26)(19,31)(25,40)(27,28)(30,45)(32,36)(33,42)(46,47);(1,4,5,3)(2,6,7,8)(17,21,22,23)(18,27,26,28)(19,32,31,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,46,45,47)(38,44,43,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6782138);