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 | |
4134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,11,16)(4,6,14,15)(5,13,12,10)(18,21)(19,25)(24,28)(26,31)(33,34,38,40)(35,43,45,39)(36,46,47,41)(37,48,44,42);(1,3)(2,8)(4,5)(6,13)(7,11)(9,16)(10,15)(12,14)(17,18,22,21)(19,23,25,27)(20,24,29,28)(26,30,31,32)(33,35,38,45)(34,39,40,43)(36,44,47,37)(41,42,46,48);(2,9)(3,11)(4,14)(10,13)(19,25)(23,27)(26,31)(30,32)(34,40)(39,43)(41,46)(42,48);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,15)(13,16)(17,19,22,25)(18,23,21,27)(20,26,29,31)(24,30,28,32)(33,34)(35,43)(36,41)(37,42)(38,40)(39,45)(44,48)(46,47);(1,5,7,12)(2,10,9,13)(3,4,11,14)(6,8,15,16)(17,20,22,29)(18,24,21,28)(19,26,25,31)(23,30,27,32)(33,36)(34,41)(35,44)(37,45)(38,47)(39,42)(40,46)(43,48);(1,6)(2,4)(3,13)(5,8)(7,15)(9,14)(10,11)(12,16)(17,21,22,18)(19,27,25,23)(20,28,29,24)(26,32,31,30)(33,37,38,44)(34,42,40,48)(35,36,45,47)(39,46,43,41)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014437);