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 | |
4169 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)(3,10,13,7)(4,9,14,15)(5,12,16,11)(17,18)(19,24)(20,22)(21,27)(23,25)(26,28)(29,32)(30,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3)(2,7)(4,12)(5,9)(6,13)(8,10)(11,14)(15,16)(17,19)(18,24)(20,30)(21,28)(22,31)(23,29)(25,32)(26,27)(35,41)(38,45)(42,47)(46,48);(2,8)(4,14)(7,10)(11,12)(18,25)(19,29)(20,28)(27,31)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,15)(10,16)(11,13)(17,20,23,28)(18,26,25,22)(19,30,29,21)(24,27,32,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,10)(4,12)(5,15)(6,13)(7,8)(9,16)(11,14)(17,21,23,30)(18,27,25,31)(19,28,29,20)(22,24,26,32)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,5,6,16)(2,11,8,12)(3,9,13,15)(4,7,14,10)(17,22,23,26)(18,28,25,20)(19,31,29,27)(21,32,30,24)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=i,e^2=gh,f^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, 9831132);