Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,16)(12,15)(13,14)(17,18,23,28)(19,24,31,38)(20,25,33,40)(21,26,22,27)(29,36,43,46)(30,37,35,42)(32,39,34,41)(44,47,45,48);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,23,31)(18,24,28,38)(20,29,33,43)(21,30,22,35)(25,36,40,46)(26,37,27,42)(32,44,34,45)(39,47,41,48);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,23,33)(18,25,28,40)(19,29,31,43)(21,34,22,32)(24,36,38,46)(26,41,27,39)(30,45,35,44)(37,48,42,47);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(20,32)(22,23)(25,39)(27,28)(29,43)(30,35)(33,34)(36,46)(37,42)(40,41);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,26,28,27)(19,30,31,35)(20,32,33,34)(24,37,38,42)(25,39,40,41)(29,44,43,45)(36,47,46,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7496141);