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 | |
4304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,14)(5,12)(6,8)(7,13)(9,15)(10,16)(33,34)(35,38)(36,41)(37,46)(39,42)(40,45)(43,47)(44,48);(1,3)(2,7)(4,10)(5,14)(6,13)(8,11)(9,16)(12,15)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(33,35,39,45)(34,40,42,38)(36,43,37,44)(41,48,46,47);(2,8)(5,16)(7,11)(9,14)(18,23)(20,29)(24,30)(26,32)(33,36,39,37)(34,41,42,46)(35,43,45,44)(38,47,40,48);(1,4,6,15)(2,9,8,14)(3,12,13,10)(5,11,16,7)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,37,39,36)(34,41,42,46)(35,44,45,43)(38,47,40,48);(1,5)(2,10)(3,9)(4,11)(6,16)(7,15)(8,12)(13,14)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,34)(35,38)(36,46)(37,41)(39,42)(40,45)(43,48)(44,47);(1,3)(2,11)(4,10)(5,9)(6,13)(7,8)(12,15)(14,16)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,38)(34,35)(36,47)(37,48)(39,40)(41,43)(42,45)(44,46)Polycyclic presentation:
2^9|b^2=ghi,c^2=gh,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8525272);