Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489394 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2965 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,21)(12,18)(13,28)(14,31)(15,20)(16,23)(17,27)(19,30)(22,35)(24,36)(25,40)(26,37)(29,38)(32,33)(34,39);(3,7)(5,8)(9,11)(10,16)(12,23)(13,29)(14,32)(15,24)(17,36)(18,33)(19,38)(20,34)(21,28)(22,27)(25,30)(26,39)(31,35)(37,40);(1,2)(3,8)(4,6)(5,7)(12,26)(13,27)(17,28)(18,37)(21,36)(22,29)(23,39)(33,40);(9,12,14,27,15,26,30,13)(10,17,19,18,20,28,31,37)(11,22,25,23,24,29,32,39)(16,33,35,36,34,40,38,21)Polycyclic presentation:
2^9|d^2=g,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396590);