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 |
| prime radical | #55043 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,18,13,16)(12,20,21,17)(19,23,22,24)(25,26)(27,29)(28,36)(30,32)(31,38)(33,35)(34,37)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,14,18)(12,19,21,22)(17,23,20,24)(25,27,30,35)(26,29,32,33)(28,34,38,40)(31,39,36,37);(1,3)(2,5)(4,8)(6,7)(12,21)(17,20)(19,22)(23,24)(25,26,30,32)(27,33,35,29)(28,31,38,36)(34,37,40,39);(3,8)(5,6)(9,12,15,21)(10,17,14,20)(11,19,13,22)(16,23,18,24)(25,26)(27,33)(28,31)(29,35)(30,32)(34,37)(36,38)(39,40);(1,2)(3,5)(4,7)(6,8)(9,13,15,11)(10,16,14,18)(12,22,21,19)(17,23,20,24)(25,28)(26,31)(27,34)(29,39)(30,38)(32,36)(33,37)(35,40);(1,3)(2,6)(4,8)(5,7)(9,14,15,10)(11,18,13,16)(12,17,21,20)(19,23,22,24)(25,29,30,33)(26,27,32,35)(28,37,38,39)(31,40,36,34)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8611283);