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 | #39825 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(13,21)(17,22)(18,24)(20,23)(25,26,32,36)(27,30,29,33)(28,35,31,34)(37,39,38,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27)(26,33)(28,38)(29,32)(30,36)(31,37)(34,40)(35,39);(10,15)(12,16)(17,22)(20,23)(25,28,32,31)(26,34,36,35)(27,37,29,38)(30,40,33,39);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,29,32,27)(26,33,36,30)(28,37,31,38)(34,40,35,39);(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,30,32,33)(26,27,36,29)(28,39,31,40)(34,38,35,37);(1,3,4,7)(2,5,6,8)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,31,32,28)(26,35,36,34)(27,37,29,38)(30,40,33,39)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621787);