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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,15)(4,13,14,8)(5,12,16,9)(18,23)(20,29)(24,27)(26,31)(33,34,40,44)(35,41,46,48)(36,37,38,42)(39,45,47,43);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,18,22,23)(19,27,28,24)(20,30,29,25)(21,26,32,31)(33,35,40,46)(34,41,44,48)(36,47,38,39)(37,43,42,45);(3,11)(4,14)(7,15)(8,13)(19,28)(21,32)(24,27)(26,31)(33,36,40,38)(34,42,44,37)(35,39,46,47)(41,43,48,45);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,19)(18,24)(20,31)(21,25)(22,28)(23,27)(26,29)(30,32)(33,37,40,42)(34,36,44,38)(35,45,46,43)(39,48,47,41);(1,5)(2,9)(3,8)(4,15)(6,16)(7,14)(10,12)(11,13)(17,20,22,29)(18,25,23,30)(19,26,28,31)(21,24,32,27)(33,38,40,36)(34,37,44,42)(35,39,46,47)(41,43,48,45);(1,3)(2,7)(4,12)(5,8)(6,11)(9,14)(10,15)(13,16)(17,21,22,32)(18,26,23,31)(19,25,28,30)(20,24,29,27)(33,39)(34,43)(35,36)(37,48)(38,46)(40,47)(41,42)(44,45)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841500);