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 | #42951 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(13,21)(17,24)(19,23)(20,22)(26,30)(27,33)(28,36)(38,40);(1,3,6,5)(2,7,4,8)(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,3,6,5)(2,8,4,7)(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4,6,2)(3,7,5,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,5,6,3)(2,7,4,8)(9,12)(10,16)(11,18)(13,20)(14,15)(17,23)(19,24)(21,22)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(9,13,14,21)(10,17,18,24)(11,19,16,23)(12,20,15,22)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8059348);