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 | |
3973 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)(3,9,10,7)(4,8,12,13)(11,15,16,14)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(2,6)(4,12)(7,9)(11,16)(18,23)(21,24)(25,28)(29,31)(34,39)(36,40)(41,45)(44,47);(3,10)(7,9)(11,16)(14,15)(18,23)(19,26)(25,28)(27,32)(33,34,38,39)(35,40,43,36)(37,41,46,45)(42,47,48,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662414);