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 | |
4016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,13,15)(4,8,14,12)(5,9,16,11)(17,18,23,21)(19,27,30,24)(20,31,29,25)(22,26,28,32)(33,34,39,44)(35,43,38,40)(36,46,45,41)(37,42,47,48);(1,3)(2,7)(4,9)(5,12)(6,13)(8,16)(10,15)(11,14)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,32)(26,31)(33,35,39,38)(34,40,44,43)(36,47,45,37)(41,48,46,42);(1,4)(2,8)(3,11)(5,7)(6,14)(9,13)(10,12)(15,16)(17,18,23,21)(19,24,30,27)(20,25,29,31)(22,26,28,32)(33,36)(34,41)(35,37)(38,47)(39,45)(40,42)(43,48)(44,46);(1,4)(2,8)(3,9)(5,15)(6,14)(7,16)(10,12)(11,13)(17,20)(18,25)(19,28)(21,31)(22,30)(23,29)(24,32)(26,27)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48);(1,3)(2,7)(4,11)(5,8)(6,13)(9,14)(10,15)(12,16)(17,21,23,18)(19,27,30,24)(20,25,29,31)(22,26,28,32)(33,37,39,47)(34,42,44,48)(35,45,38,36)(40,46,43,41);(1,5)(2,9)(3,12)(4,7)(6,16)(8,13)(10,11)(14,15)(17,22,23,28)(18,26,21,32)(19,29,30,20)(24,31,27,25)(33,38,39,35)(34,43,44,40)(36,47,45,37)(41,48,46,42)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9121926);