Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,21,36,22,27,23,26)(19,28,37,40,31,24,30,39)(20,32,33,25,34,42,35,41)(29,44,47,43,46,48,45,38);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,22,31)(18,24,27,28)(20,29,34,46)(21,37,23,30)(25,38,41,43)(26,40,36,39)(32,48,42,44)(33,47,35,45);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,34)(18,25,27,41)(19,29,31,46)(21,35,23,33)(24,38,28,43)(26,42,36,32)(30,47,37,45)(39,48,40,44);(2,6)(5,7)(10,13)(14,16)(17,21,22,23)(18,26,27,36)(19,30,31,37)(20,33,34,35)(24,39,28,40)(25,32,41,42)(29,45,46,47)(38,44,43,48);(9,11,12,15)(10,14,13,16)(19,31)(24,28)(29,46)(30,37)(38,43)(39,40)(44,48)(45,47)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=i,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6873728);