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 | |
4099 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)(4,14)(5,9)(6,7)(8,11)(12,13)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(2,7)(3,11)(4,9)(13,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34,39,38)(35,42,36,40)(37,45,46,41)(43,47,44,48);(2,7)(5,14)(8,10)(13,16)(20,28)(23,30)(26,32)(29,31)(33,35,39,36)(34,40,38,42)(37,43,46,44)(41,47,45,48);(1,3)(2,8)(4,12)(5,13)(6,11)(7,10)(9,15)(14,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,39,35)(34,40,38,42)(37,44,46,43)(41,47,45,48);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,37,39,46)(34,41,38,45)(35,43,36,44)(40,47,42,48);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,38,39,34)(35,40,36,42)(37,45,46,41)(43,47,44,48)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8177811);