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 | |
4212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,10,4)(5,7,13,14)(11,15,16,12)(17,18)(19,27)(20,31)(21,26)(22,23)(24,30)(25,29)(28,32)(33,34)(35,40)(36,42)(37,38)(39,44)(41,45)(43,48)(46,47);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19,23,30)(18,24,22,27)(20,28,29,21)(25,32,31,26)(33,35,39,46)(34,40,44,47)(36,43,45,37)(38,42,48,41);(3,10)(4,9)(5,13)(7,14)(18,22)(20,29)(21,28)(24,27)(33,36,39,45)(34,41,44,42)(35,37,46,43)(38,40,48,47);(1,3,6,10)(2,4,8,9)(5,11,13,16)(7,12,14,15)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,32)(26,27)(33,36)(34,42)(35,43)(37,46)(38,47)(39,45)(40,48)(41,44);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,31,27,25)(33,37,39,43)(34,38,44,48)(35,45,46,36)(40,41,47,42);(1,5,6,13)(2,7,8,14)(3,11,10,16)(4,12,9,15)(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,32,28,26)(33,38)(34,43)(35,42)(36,47)(37,44)(39,48)(40,45)(41,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=i,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900169);