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 | |
4022 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)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(33,34,39,43)(35,37,44,40)(36,41,45,46)(38,42,47,48);(2,7)(3,11)(4,14)(13,15)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(35,44)(36,45)(37,40)(41,46);(2,7)(4,14)(8,10)(12,16)(18,23)(20,29)(24,30)(26,32)(33,35,39,44)(34,40,43,37)(36,38,45,47)(41,48,46,42);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,36)(34,41)(35,38)(37,42)(39,45)(40,48)(43,46)(44,47);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,37,39,40)(34,35,43,44)(36,42,45,48)(38,46,47,41);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,38)(34,42)(35,45)(36,44)(37,46)(39,47)(40,41)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8525339);