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 | |
4405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,23)(18,22)(21,24)(26,31)(28,35)(32,38)(34,37);(1,2,4,5)(3,6,7,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,26)(27,32)(28,36)(29,38)(30,31)(33,35)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,19)(17,21,24,23)(25,27,30,29)(26,32,31,38)(28,37,35,34)(33,40,36,39);(2,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28,30,35)(26,33,31,36)(27,34,29,37)(32,39,38,40);(1,2)(3,8)(4,5)(6,7)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,29,30,27)(26,32,31,38)(28,34,35,37)(33,40,36,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)(25,26)(27,32)(28,33)(29,38)(30,31)(34,39)(35,36)(37,40)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=g,e^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, 10219844);