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 | |
3952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,8,11)(5,9,14,15)(10,12,16,13)(17,18,22,25)(19,21,27,23)(20,24,28,31)(26,29,32,30)(33,34,38,42)(35,39,44,48)(36,45,46,40)(37,41,43,47);(2,7)(4,11)(9,15)(12,13)(20,28)(24,31)(26,32)(29,30)(33,35,38,44)(34,39,42,48)(36,43,46,37)(40,41,45,47);(17,19,22,27)(18,23,25,21)(20,26,28,32)(24,30,31,29)(33,36,38,46)(34,40,42,45)(35,37,44,43)(39,47,48,41);(1,3,6,8)(2,4,7,11)(5,10,14,16)(9,12,15,13)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,32)(30,31)(33,36)(34,40)(35,43)(37,44)(38,46)(39,41)(42,45)(47,48);(1,4)(2,8)(3,7)(5,13)(6,11)(9,10)(12,14)(15,16)(17,20,22,28)(18,24,25,31)(19,26,27,32)(21,29,23,30)(33,37,38,43)(34,41,42,47)(35,36,44,46)(39,45,48,40);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,21)(18,19)(20,29)(22,23)(24,26)(25,27)(28,30)(31,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,48)(45,46)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820851);