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 | |
4130 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)(7,12)(8,13)(14,16)(17,18)(19,21)(20,28)(22,24)(23,27)(25,30)(26,31)(29,32)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(1,2,5,6)(3,7,10,12)(4,8,11,13)(9,14,15,16)(17,19)(18,23)(20,26)(21,24)(22,27)(25,31)(28,29)(30,32)(33,34,38,39)(35,37,41,46)(36,40,45,43)(42,44,47,48);(4,11)(8,13)(9,15)(14,16)(18,24)(21,23)(25,28)(29,31)(34,39)(37,46)(40,43)(44,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20,22,30)(18,25,24,28)(19,26,27,32)(21,29,23,31)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,19)(18,23)(20,26)(21,24)(22,27)(25,31)(28,29)(30,32)(33,36,38,45)(34,40,39,43)(35,42,41,47)(37,44,46,48);(1,4,5,11)(2,8,6,13)(3,9,10,15)(7,14,12,16)(17,21,22,23)(18,19,24,27)(20,29,30,31)(25,26,28,32)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8086829);