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 | #45045 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4225 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,17)(13,18)(14,19)(16,20)(21,23)(22,24)(25,26)(27,32)(28,29)(30,40)(31,34)(33,38)(35,39)(36,37);(1,3,6,5)(2,7,4,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,19)(17,23)(18,24)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(1,3,6,5)(2,7,4,8)(11,20)(15,16)(21,24)(22,23)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(3,5)(7,8)(9,11)(10,16)(12,21)(13,23)(14,20)(15,19)(17,22)(18,24)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(1,4,6,2)(3,8,5,7)(9,12)(10,17)(11,21)(13,19)(14,18)(15,23)(16,22)(20,24)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36);(1,5,6,3)(2,7,4,8)(9,13,14,17)(10,18,19,12)(11,22,20,23)(15,24,16,21)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653793);