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 | C2×#535 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324 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)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(26,31)(27,29)(33,36)(35,37);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,26,30,31)(27,32,29,34)(28,36,38,33)(35,40,37,39);(1,3,5,8)(2,6,4,7)(10,14)(11,13)(12,20)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(3,8)(6,7)(9,12,15,20)(10,17,14,21)(11,19,13,22)(16,23,18,24)(25,27,30,29)(26,32,31,34)(28,35,38,37)(33,39,36,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,11)(10,18,14,16)(12,19,20,22)(17,23,21,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,4,5,2)(3,6,8,7)(9,14,15,10)(11,18,13,16)(12,21,20,17)(19,24,22,23)(25,29,30,27)(26,34,31,32)(28,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=hi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8936146);