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 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,25)(20,29)(21,30)(22,24)(23,28)(26,32)(27,31)(33,34,39,36)(35,42,40,37)(38,44,46,41)(43,45,47,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,19,22,28)(18,23,24,25)(20,26,30,31)(21,32,29,27)(35,40)(37,42)(38,46)(41,44);(2,6)(7,10)(8,12)(14,16)(18,24)(20,30)(23,25)(26,31)(33,35,39,40)(34,37,36,42)(38,43,46,47)(41,45,44,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,21)(19,26)(22,30)(23,32)(24,29)(25,27)(28,31)(33,36,39,34)(35,37,40,42)(38,44,46,41)(43,45,47,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,19)(18,23)(20,31)(21,27)(22,28)(24,25)(26,30)(29,32)(33,37)(34,40)(35,36)(38,45)(39,42)(41,47)(43,44)(46,48);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,38,39,46)(34,41,36,44)(35,43,40,47)(37,45,42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8782812);