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 | |
4198 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)(3,4)(5,13)(6,8)(7,9)(10,14)(11,15)(12,16)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30)(33,34)(35,40)(36,42)(37,38)(39,44)(41,45)(43,48)(46,47);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,39,46)(34,40,44,47)(36,43,45,37)(38,42,48,41);(2,8)(4,7)(5,14)(11,16)(19,24)(20,23)(21,29)(25,32)(33,36,39,45)(34,41,44,42)(35,37,46,43)(38,40,48,47);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,19)(18,23)(20,26)(21,27)(22,24)(25,30)(28,32)(29,31)(33,36)(34,42)(35,43)(37,46)(38,47)(39,45)(40,48)(41,44);(1,2)(3,4)(5,10)(6,8)(7,9)(11,12)(13,14)(15,16)(17,20,22,23)(18,24,26,19)(21,28,29,30)(25,31,32,27)(33,37,39,43)(34,38,44,48)(35,45,46,36)(40,41,47,42);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(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=hi,e^2=gh,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, 8900177);