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 | |
4327 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)(3,8)(4,6)(5,11)(7,9)(10,12)(13,16)(14,15)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,31)(29,30)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3)(2,8)(4,14)(5,12)(6,15)(7,13)(9,16)(10,11)(17,19,22,21)(18,23,24,26)(20,27,28,29)(25,30,31,32)(35,41)(38,45)(42,47)(46,48);(2,9)(3,13)(4,12)(11,15)(18,24)(23,26)(25,31)(30,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4,7,12)(2,10,9,6)(3,14,13,5)(8,11,16,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5,7,14)(2,11,9,15)(3,12,13,4)(6,8,10,16)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,6,7,10)(2,12,9,4)(3,15,13,11)(5,16,14,8)(17,21,22,19)(18,26,24,23)(20,27,28,29)(25,30,31,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)Polycyclic presentation:
2^9|b^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, 9831133);