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 | |
4010 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)(3,9)(4,11)(5,13)(6,7)(8,14)(10,12)(15,16)(21,29)(25,30)(26,31)(27,32)(34,36)(39,41)(40,44)(46,47);(1,3,6,11)(2,4,7,9)(5,10,14,16)(8,12,13,15)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,7)(4,9)(8,13)(12,15)(18,23)(19,24)(25,30)(26,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,16)(14,15)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32);(1,5,6,14)(2,8,7,13)(3,10,11,16)(4,12,9,15)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019751);