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 | #2670 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(10,11)(13,21)(15,17)(19,24)(25,26,30,34)(27,35,38,31)(28,32,29,33)(36,40,37,39);(1,2)(3,6)(4,7)(5,8)(25,27,30,38)(26,31,34,35)(28,36,29,37)(32,39,33,40);(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,40)(34,35);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,35,40);(1,4,5,3)(2,7,8,6)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,28)(26,33,34,32)(27,37,38,36)(31,40,35,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7553061);