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 | |
4512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(12,19)(16,21)(17,23)(22,24)(26,31)(28,35)(29,37)(34,39);(1,2,4,5)(3,6,7,8)(10,14)(15,20)(16,21)(22,24)(26,31)(27,33)(28,35)(36,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27,30,33)(26,29,31,37)(28,34,35,39)(32,36,38,40);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,29)(26,33)(27,31)(28,36)(30,37)(32,39)(34,38)(35,40)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8016367);