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 | |
4618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(10,14)(15,17)(16,20)(22,23)(25,26)(27,32)(28,34)(29,38)(30,36)(31,33)(35,37)(39,40);(1,3,6,5)(2,7,8,4)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27)(26,32)(28,37)(29,39)(30,33)(31,36)(34,35)(38,40);(2,8)(4,7)(11,18)(15,17)(19,24)(22,23)(26,33)(27,36)(29,37)(34,40);(2,8)(3,5)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,28)(26,34)(27,29)(30,35)(31,39)(32,38)(33,40)(36,37);(1,4,6,7)(2,3,8,5)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,29)(26,35)(27,28)(30,34)(31,37)(32,40)(33,38)(36,39);(1,5,6,3)(2,7,8,4)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30)(26,36)(27,33)(28,38)(29,34)(31,32)(35,39)(37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528563);