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 | #56380 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4337 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)(5,7)(6,8)(9,10,15,19)(11,14,18,23)(12,22,21,16)(13,17,20,24)(26,31)(27,34)(28,36)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,18)(10,14,19,23)(12,13,21,20)(16,24,22,17)(25,26)(27,32)(28,37)(29,34)(30,31)(33,36)(35,39)(38,40);(9,12,15,21)(10,16,19,22)(11,20,18,13)(14,17,23,24)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,38,37,39);(2,7)(4,6)(9,11)(10,14)(12,20)(13,21)(15,18)(16,17)(19,23)(22,24)(25,28)(26,33)(27,35)(29,39)(30,36)(31,37)(32,38)(34,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,20)(10,17,19,24)(11,21,18,12)(14,16,23,22);(1,4,5,6)(2,3,7,8)(9,14)(10,18)(11,19)(12,17)(13,16)(15,23)(20,22)(21,24)(25,29,30,32)(26,34,31,27)(28,38,36,39)(33,35,37,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8352869);