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 | |
4179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(5,13)(12,15)(17,18)(19,24)(20,21)(22,26)(23,29)(25,31)(27,32)(28,30)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(1,2,6,7)(3,4,9,8)(5,10,13,14)(11,12,16,15)(17,18,22,26)(19,23,29,24)(20,21,25,31)(27,28,30,32)(34,36)(40,41)(42,44)(47,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,30)(31,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3)(2,4)(5,11)(6,9)(7,8)(10,12)(13,16)(14,15)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32)(33,36,39,34)(35,41,37,40)(38,44,46,42)(43,48,45,47);(1,4)(2,9)(3,7)(5,12)(6,8)(10,16)(11,14)(13,15)(17,20)(18,21)(19,27)(22,25)(23,28)(24,32)(26,31)(29,30)(33,37,39,35)(34,41,36,40)(38,43,46,45)(42,47,44,48);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,21)(18,25)(19,28)(20,26)(22,31)(23,30)(24,27)(29,32)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10362914);