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 | |
4485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26)(27,33)(28,29)(30,34)(31,38)(32,39)(35,40)(36,37);(1,2,4,5)(3,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,27)(26,31)(28,35)(29,37)(30,38)(32,40)(33,34)(36,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,28)(26,32)(27,35)(29,34)(30,39)(31,40)(33,37)(36,38);(2,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28)(26,32)(27,36)(29,34)(30,39)(31,37)(33,40)(35,38);(1,2)(3,8)(4,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,33)(28,35)(29,40)(30,38)(31,34)(32,37)(36,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,29,30,32)(26,28,34,39)(27,37,38,40)(31,35,33,36)Polycyclic presentation:
2^9|b^2=i,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10219765);