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 | #40730 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,23)(13,24)(15,17)(16,22)(18,21)(19,20);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,17,14)(12,20,21,13)(18,19,23,24)(25,26,30,31)(27,33,36,32)(28,37,39,29)(34,40,38,35);(1,3,5,8)(2,6,4,7)(10,17)(14,16)(18,23)(19,24)(26,31)(29,37)(32,33)(35,40);(2,4)(6,7)(9,12)(10,18)(11,20)(13,22)(14,24)(15,21)(16,19)(17,23)(25,27,30,36)(26,32,31,33)(28,38,39,34)(29,35,37,40);(1,4,5,2)(3,7,8,6)(9,13,15,20)(10,19,17,24)(11,21,22,12)(14,23,16,18)(25,28)(26,29)(27,34)(30,39)(31,37)(32,40)(33,35)(36,38);(1,3)(2,6)(4,7)(5,8)(9,14)(10,11)(12,24)(13,23)(15,16)(17,22)(18,20)(19,21)(25,29)(26,28)(27,35)(30,37)(31,39)(32,38)(33,34)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300516);