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 | |
3979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,13,16)(4,10,12,6)(5,15,14,11)(17,18)(19,21)(20,28)(22,23)(24,27)(25,29)(26,31)(30,32)(34,37)(39,42)(40,45)(46,48);(1,3)(2,8)(4,14)(5,12)(6,15)(7,13)(9,16)(10,11)(18,23)(20,29)(21,24)(26,32)(36,44)(40,45)(41,47)(46,48);(2,9)(3,13)(4,12)(11,15)(17,19,22,27)(18,21,23,24)(20,26,29,32)(25,30,28,31)(33,34,38,37)(35,39,43,42)(36,40,44,45)(41,46,47,48);(1,4,7,12)(2,10,9,6)(3,14,13,5)(8,11,16,15)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,30)(29,32)(33,35,38,43)(34,39,37,42)(36,41,44,47)(40,46,45,48);(1,5,7,14)(2,11,9,15)(3,12,13,4)(6,8,10,16)(17,20,22,29)(18,25,23,28)(19,26,27,32)(21,30,24,31)(33,36)(34,40)(35,41)(37,45)(38,44)(39,46)(42,48)(43,47);(1,6,7,10)(2,12,9,4)(3,11,13,15)(5,8,14,16)(17,21,22,24)(18,19,23,27)(20,30,29,31)(25,26,28,32)(33,37,38,34)(35,42,43,39)(36,45,44,40)(41,48,47,46)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664568);