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 | |
4516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,26,30,35)(27,34,37,31)(28,38,36,32)(29,33,39,40);(3,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,27,30,37)(26,31,35,34)(28,29,36,39)(32,33,38,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,17,15)(12,18,21,24)(16,22,20,23)(25,28,30,36)(26,32,35,38)(27,29,37,39)(31,33,34,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,29,30,39)(26,33,35,40)(27,36,37,28)(31,38,34,32);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,20)(13,17)(15,19)(16,21)(18,23)(22,24)(25,27)(26,34)(28,29)(30,37)(31,35)(32,40)(33,38)(36,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,20)(19,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9089046);