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 | |
4066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,12,14,8)(11,16,13,15)(17,18,22,23)(19,21,26,24)(20,25,30,32)(27,31,28,29)(33,34,39,41)(35,38,36,40)(37,42,46,48)(43,45,44,47);(2,6)(3,9)(8,12)(11,13)(18,23)(19,26)(25,32)(27,28)(33,35,39,36)(34,40,41,38)(37,44,46,43)(42,45,48,47);(2,6)(7,10)(8,12)(15,16)(34,41)(35,36)(42,48)(43,44);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,14)(13,15)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,31,32,29)(33,36,39,35)(34,38,41,40)(37,44,46,43)(42,45,48,47);(1,3,5,9)(2,7,6,10)(4,13,14,11)(8,16,12,15)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,37,39,46)(34,42,41,48)(35,43,36,44)(38,45,40,47);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,13)(10,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,32,28)(33,38)(34,35)(36,41)(37,45)(39,40)(42,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=ghi,e^2=g,f^2=gh,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, 10154872);