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 | |
4363 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,19)(13,21)(14,16)(17,22)(18,24)(20,23)(26,31)(29,32)(33,38)(37,39);(2,5)(6,7)(9,11,14,12)(10,15,16,19)(13,18,22,20)(17,23,21,24)(28,36)(33,38)(35,40)(37,39);(1,2)(3,6)(4,5)(7,8)(10,16)(11,12)(13,22)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,15,16,19)(13,20,22,18)(17,23,21,24)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(9,10)(11,15)(12,19)(13,17)(14,16)(18,23)(20,24)(21,22)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(9,13,14,22)(10,17,16,21)(11,18,12,20)(15,23,19,24)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|b^2=gi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961610);