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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,22,23)(19,21,26,24)(20,25,30,32)(27,31,28,29)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,47)(46,48);(4,11)(7,9)(12,16)(14,15)(18,23)(19,26)(25,32)(27,28)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,3)(7,9)(8,10)(14,15)(34,35)(39,41)(40,42)(46,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,31,32,29)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,32,28)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154327);