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 | |
4226 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,14)(15,16)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(34,38)(36,44)(39,41)(42,48);(2,7)(5,8)(9,14)(13,15)(17,19,23,21)(18,24,20,25)(22,29,30,27)(26,32,28,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(4,12)(9,14)(10,16)(13,15)(19,21)(24,25)(27,29)(31,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,6,12)(2,9,7,14)(3,10,11,16)(5,13,8,15)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,6,8)(2,3,7,11)(4,13,12,15)(9,10,14,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=h,d^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9275338);