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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3999 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)(3,7)(4,5)(6,8)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(26,32)(27,35)(28,38)(30,40);(2,5)(6,7)(11,19)(13,23)(15,24)(17,20)(25,26,31,32)(27,33,35,29)(28,34,38,37)(30,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,27)(26,33)(28,39)(29,32)(30,37)(31,35)(34,40)(36,38);(1,3,4,8)(2,6,5,7)(25,28)(26,34)(27,36)(29,40)(30,33)(31,38)(32,37)(35,39);(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,29,31,33)(26,35,32,27)(28,40,38,30)(34,39,37,36);(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)(25,30,31,40)(26,36,32,39)(27,37,35,34)(28,33,38,29)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9148131);