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 | |
4163 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)(3,8)(4,14)(5,15)(6,9)(7,13)(10,16)(11,12)(17,18)(19,21)(20,25)(22,27)(23,26)(24,31)(28,32)(29,30)(33,34,39,36)(35,41,37,40)(38,42,46,44)(43,48,45,47);(1,3)(2,7)(4,11)(5,12)(6,13)(8,9)(10,15)(14,16)(17,19)(18,23)(20,28)(21,27)(22,26)(24,32)(25,30)(29,31)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,4)(2,5)(3,10)(6,15)(7,16)(8,12)(9,14)(11,13)(17,20)(18,24)(19,28)(21,30)(22,31)(23,32)(25,27)(26,29)(33,35,39,37)(34,41,36,40)(38,43,46,45)(42,48,44,47);(1,4)(2,5)(3,11)(6,15)(7,12)(8,16)(9,14)(10,13)(17,20)(18,25)(19,29)(21,30)(22,31)(23,32)(24,27)(26,28)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,3)(2,8)(4,10)(5,12)(6,13)(7,9)(11,15)(14,16)(17,19)(18,21)(20,28)(22,26)(23,27)(24,30)(25,32)(29,31)(33,37,39,35)(34,40,36,41)(38,43,46,45)(42,48,44,47);(1,5)(2,4)(3,12)(6,14)(7,11)(8,10)(9,15)(13,16)(17,21)(18,26)(19,27)(20,30)(22,23)(24,29)(25,28)(31,32)(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=gh,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, 10362916);