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 | |
4235 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)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18)(19,24)(20,21)(22,23)(25,30)(26,29)(27,31)(28,32)(33,34)(35,46)(36,43)(37,42)(38,47)(39,41)(40,45)(44,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,28)(31,32)(33,35,40,39)(34,41,45,46)(36,47,37,44)(38,43,48,42);(5,14)(8,12)(10,13)(15,16)(18,23)(20,30)(24,26)(27,32)(33,36,40,37)(34,42,45,43)(35,47,39,44)(38,46,48,41);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,19)(18,24)(20,27)(21,31)(22,29)(23,26)(25,28)(30,32)(33,37,40,36)(34,43,45,42)(35,47,39,44)(38,46,48,41);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,20)(18,25)(19,27)(21,23)(22,30)(24,28)(26,31)(29,32)(33,38,40,48)(34,44,45,47)(35,42,39,43)(36,41,37,46);(1,3)(2,9)(4,7)(5,10)(6,11)(8,16)(12,15)(13,14)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,32,26,27)(33,39,40,35)(34,41,45,46)(36,44,37,47)(38,43,48,42)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,d^2=g,e^2=g,f^2=h,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, 8514639);