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 | |
4280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,14)(4,5,12,13)(9,10,15,16)(17,18,22,24)(19,23,27,30)(20,25,29,32)(21,28,31,26)(33,34)(35,45)(36,39)(37,42)(38,43)(40,44)(41,47)(46,48);(1,3,6,11)(2,7,8,14)(4,9,12,15)(5,10,13,16)(17,19,22,27)(18,23,24,30)(20,28,29,26)(21,25,31,32)(33,35,40,47)(34,41,44,45)(36,46,37,38)(39,43,42,48);(2,8)(4,12)(7,14)(9,15)(18,24)(19,27)(25,32)(26,28)(33,36,40,37)(34,39,44,42)(35,46,47,38)(41,43,45,48);(1,4)(2,5)(3,9)(6,12)(7,10)(8,13)(11,15)(14,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32)(33,37,40,36)(34,42,44,39)(35,38,47,46)(41,48,45,43);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20,22,29)(18,25,24,32)(19,26,27,28)(21,30,31,23)(33,38)(34,43)(35,36)(37,47)(39,45)(40,46)(41,42)(44,48);(1,5)(2,4)(3,10)(6,13)(7,9)(8,12)(11,16)(14,15)(17,21)(18,26)(19,25)(20,23)(22,31)(24,28)(27,32)(29,30)(33,39,40,42)(34,36,44,37)(35,43,47,48)(38,41,46,45)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9624585);