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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4468 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)(3,5)(4,7)(6,8)(9,10,15,17)(11,16,13,19)(12,18,23,22)(14,21,24,20);(1,2,6,8)(3,5,7,4)(9,11)(10,16)(12,21)(13,15)(14,22)(17,19)(18,24)(20,23)(25,26,30,31)(27,32,35,40)(28,33,38,37)(29,34,39,36);(1,3,6,7)(2,4,8,5)(10,17)(11,13)(12,23)(14,24)(26,31)(27,35)(28,38)(29,39);(1,4,6,5)(2,3,8,7)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,16,24,19)(25,27)(26,32)(28,34)(29,37)(30,35)(31,40)(33,39)(36,38);(1,2)(3,4)(5,7)(6,8)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,18,24,22)(25,28)(26,33)(27,36)(29,32)(30,38)(31,37)(34,35)(39,40);(1,5,6,4)(2,7,8,3)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,16,23,19)(25,29)(26,34)(27,37)(28,32)(30,39)(31,36)(33,35)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8886951);