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 | #42975 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,4)(9,10,15,17)(11,16,19,24)(12,14,22,23)(13,21,20,18)(26,29)(27,28)(34,37)(35,36);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,13)(14,21)(15,19)(17,24)(18,23)(20,22)(25,26,31,29)(27,33,28,32)(30,34,38,37)(35,40,36,39);(1,2,5,6)(3,7,4,8)(10,17)(11,19)(12,22)(18,21)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,20)(13,19)(15,22)(16,18)(17,23)(21,24)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(1,3,5,4)(2,7,6,8)(9,13,15,20)(10,18,17,21)(11,12,19,22)(14,16,23,24)(25,29,31,26)(27,33,28,32)(30,37,38,34)(35,40,36,39);(1,4,5,3)(2,8,6,7)(9,14)(10,12)(11,21)(13,16)(15,23)(17,22)(18,19)(20,24)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014431);