Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343678 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | #62838 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #3444 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,22,28,23,27,21,26)(19,29,31,42,32,24,36,38)(20,25,35,41,34,40,33,39)(30,43,44,48,45,37,46,47);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,38)(27,29)(28,42)(33,44)(34,45)(35,46)(39,47)(40,43)(41,48);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(17,20)(18,25)(19,30)(21,33)(22,35)(23,34)(24,37)(26,39)(27,40)(28,41)(29,43)(31,44)(32,45)(36,46)(38,47)(42,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,27,28)(19,31,32,36)(20,33,34,35)(24,38,29,42)(25,39,40,41)(30,44,45,46)(37,47,43,48);(2,9)(7,16)(8,13)(10,15)(20,34)(25,40)(30,45)(33,35)(37,43)(39,41)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,f^2=i,g^2=hi,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251479);