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 | |
4057 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)(3,8)(4,9)(5,14)(6,7)(10,16)(11,15)(12,13)(17,18)(19,27)(20,25)(21,31)(22,23)(24,29)(26,32)(28,30)(33,34,39,42)(35,40,38,36)(37,41,47,48)(43,45,46,44);(1,2)(3,8)(4,12)(5,10)(6,7)(9,13)(11,15)(14,16)(18,23)(20,30)(24,27)(26,31)(33,35,39,38)(34,40,42,36)(37,43,47,46)(41,45,48,44);(2,7)(3,11)(4,13)(5,16)(17,19,22,29)(18,24,23,27)(20,31,30,26)(21,25,32,28)(33,36,39,40)(34,35,42,38)(37,45,47,44)(41,46,48,43);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,19)(18,24)(20,31)(21,28)(22,29)(23,27)(25,32)(26,30)(33,36)(34,35)(37,45)(38,42)(39,40)(41,46)(43,48)(44,47);(1,4)(2,9)(3,10)(5,15)(6,13)(7,12)(8,16)(11,14)(17,20,22,30)(18,25,23,28)(19,26,29,31)(21,27,32,24)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,21)(18,26)(19,28)(20,27)(22,32)(23,31)(24,30)(25,29)(33,38,39,35)(34,36,42,40)(37,46,47,43)(41,44,48,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8936771);