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 | |
4001 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)(4,8)(5,7)(6,9)(11,15)(12,14)(13,16)(17,18,22,26)(19,27,29,23)(20,30,28,24)(21,25,31,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3,6,5)(2,7,9,10)(4,12,13,11)(8,15,16,14)(17,19)(18,23)(20,31)(21,28)(22,29)(24,32)(25,30)(26,27)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(17,20,22,28)(18,24,26,30)(19,21,29,31)(23,25,27,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4,6,13)(2,8,9,16)(3,11,5,12)(7,14,10,15)(17,19)(18,23)(20,21)(22,29)(24,25)(26,27)(28,31)(30,32)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,21)(18,25)(19,28)(20,29)(22,31)(23,30)(24,27)(26,32)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,16,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,30)(31,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=hi,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, 10411680);