Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(10,14)(11,12)(17,18,22,26)(19,29,31,24)(20,33,35,25)(21,27,32,42)(23,28,38,43)(30,39,46,44)(34,40,36,47)(37,48,45,41);(9,10)(11,16)(12,15)(13,14)(17,19,22,31)(18,24,26,29)(20,34,35,36)(21,38,32,23)(25,40,33,47)(27,28,42,43)(30,45,46,37)(39,48,44,41);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,23,37,22,35,38,45)(18,25,28,41,26,33,43,48)(19,30,32,34,31,46,21,36)(24,39,27,40,29,44,42,47);(9,12,13,11)(10,15,14,16)(18,26)(20,35)(24,29)(27,42)(28,43)(30,46)(34,36)(37,45);(1,3,4,7)(2,6,5,8)(17,21,38,31,22,32,23,19)(18,27,43,24,26,42,28,29)(20,36,45,46,35,34,37,30)(25,40,48,39,33,47,41,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=h,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6740604);