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 | |
4175 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)(3,10)(4,9)(5,14)(6,8)(7,13)(11,12)(15,16)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,34,39,37)(35,40,36,41)(38,42,46,45)(43,47,44,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,15)(14,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(34,37)(40,41)(42,45)(47,48);(2,8)(4,14)(7,10)(11,16)(18,23)(19,25)(24,31)(26,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,4)(2,9)(3,11)(5,8)(6,14)(7,15)(10,12)(13,16)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,32)(28,31)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,3)(2,10)(4,11)(5,15)(6,13)(7,8)(9,12)(14,16)(17,20,22,28)(18,24,23,31)(19,26,25,32)(21,29,30,27)(33,37,39,34)(35,41,36,40)(38,42,46,45)(43,47,44,48);(1,5,6,9)(2,4,8,14)(3,12,13,15)(7,11,10,16)(17,21)(18,19)(20,27)(22,30)(23,25)(24,32)(26,31)(28,29)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9830844);