Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,12,13,7)(10,15,16,14)(17,18,23,28)(19,26,21,24)(20,30,31,25)(22,27,29,32)(35,38)(40,42)(43,45)(47,48);(1,2,5,8)(3,6,11,9)(4,12,13,7)(10,15,16,14)(17,19,23,21)(18,24,28,26)(20,22,31,29)(25,27,30,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,20,23,31)(18,25,28,30)(19,29,21,22)(24,32,26,27)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,21,23,19)(18,26,28,24)(20,22,31,29)(25,27,30,32)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,22)(18,27)(19,20)(21,31)(23,29)(24,25)(26,30)(28,32)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,32)(30,31)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8780795);