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 | |
4163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(9,10)(11,20)(12,18)(13,19)(14,16)(15,17)(21,24)(22,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,39)(37,40);(1,3,5,7)(2,4,8,6)(9,11,15,14)(10,16,17,20)(12,21,22,19)(13,23,24,18)(25,27)(26,31)(28,36)(29,37)(30,38)(32,33)(34,35)(39,40);(1,3)(2,6)(4,8)(5,7)(10,17)(12,22)(16,20)(19,21)(25,28,30,37)(26,32,34,40)(27,36,38,29)(31,33,35,39);(1,2,5,8)(3,4,7,6)(9,12,15,22)(10,18,17,23)(11,21,14,19)(13,20,24,16);(1,3)(2,4)(5,7)(6,8)(9,13)(10,19)(11,18)(12,20)(14,23)(15,24)(16,22)(17,21)(25,29,30,36)(26,33,34,39)(27,37,38,28)(31,32,35,40);(1,4)(2,7)(3,8)(5,6)(9,14,15,11)(10,20,17,16)(12,19,22,21)(13,18,24,23)(25,26)(27,31)(28,32)(29,39)(30,34)(33,36)(35,38)(37,40)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=h,e^2=ghi,f^2=hi,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, 9633346);