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 | #45610 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,7,5)(9,10)(11,17)(12,13)(14,23)(15,16)(18,22)(19,20)(21,24)(25,26,31,34)(27,30,37,32)(28,33,38,29)(35,36,39,40);(2,4)(3,7)(10,16)(13,18)(17,24)(19,23)(28,38)(29,33)(35,39)(36,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,21)(10,17,16,24)(12,20,22,14)(13,19,18,23)(25,27,31,37)(26,32,34,30)(28,35,38,39)(29,36,33,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,20)(13,16)(14,21)(15,22)(17,23)(19,24)(25,28)(26,33)(27,35)(29,34)(30,36)(31,38)(32,40)(37,39);(1,4,6,2)(3,8,7,5)(9,13,15,18)(10,12,16,22)(11,19,21,23)(14,17,20,24)(25,29,31,33)(26,28,34,38)(27,36,37,40)(30,39,32,35);(1,5,6,8)(2,7,4,3)(9,14,15,20)(10,19,16,23)(11,12,21,22)(13,17,18,24)(25,30)(26,27)(28,36)(29,39)(31,32)(33,35)(34,37)(38,40)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000519);