Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500019 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62823 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18)(19,27,30,23)(20,24)(21,34)(22,26)(25,36)(28,37,31,39)(29,38,35,42)(32,40)(33,46)(41,45)(43,48,44,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,35)(22,30)(24,39)(25,42)(26,27)(28,32)(29,36)(33,43)(34,38)(37,40)(41,47)(44,45)(46,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,22,32)(18,24,26,40)(19,28,30,31)(21,33,36,45)(23,37,27,39)(25,41,34,46)(29,43,35,44)(38,47,42,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(20,32)(24,40)(28,31)(33,45)(37,39)(41,46)(43,44)(47,48);(2,8)(6,9)(7,16)(14,15)(17,21,22,36)(18,25,26,34)(19,29,30,35)(20,33,32,45)(23,38,27,42)(24,41,40,46)(28,43,31,44)(37,47,39,48)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=hi,e^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6438239);