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 | |
4117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,11)(10,15)(17,18)(19,25)(20,23)(21,24)(22,28)(26,30)(27,31)(29,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(1,2,5,6)(3,7,9,13)(4,11,12,8)(10,16,15,14)(17,19,23,21)(18,24,20,25)(22,29,30,27)(26,32,28,31)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(3,9)(7,13)(10,15)(14,16)(18,20)(24,25)(26,28)(31,32)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=hi,e^2=h,f^2=g,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, 8662375);