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 | #44038 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(2,7)(3,12)(9,14)(10,13)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(2,7)(3,12)(9,14)(10,13)(20,29)(24,28)(27,30)(31,32)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,16)(10,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21,22,19)(18,23,25,26)(20,30,29,27)(24,31,28,32)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,18)(19,23)(20,28)(21,26)(22,25)(24,29)(27,32)(30,31)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=i,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627873);