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 | |
4068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18,22,26)(19,27,29,23)(20,30,31,24)(21,25,28,32)(33,34,39,38)(35,42,40,36)(37,45,46,41)(43,44,47,48);(2,6)(7,9)(8,12)(14,15)(17,19,22,29)(18,23,26,27)(20,28,31,21)(24,32,30,25)(33,35,39,40)(34,36,38,42)(37,43,46,47)(41,44,45,48);(2,6)(3,10)(8,12)(11,16)(20,31)(21,28)(24,30)(25,32)(33,36,39,42)(34,40,38,35)(37,44,46,48)(41,47,45,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,28)(21,29)(22,31)(23,32)(25,27)(26,30)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,21,22,28)(18,25,26,32)(19,20,29,31)(23,24,27,30)(33,37,39,46)(34,41,38,45)(35,43,40,47)(36,44,42,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,32)(30,31)(33,38,39,34)(35,42,40,36)(37,45,46,41)(43,48,47,44)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034797);