Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488895 |
| 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 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,16,23)(11,24,29,39)(12,22,15,19)(13,17,30,21)(14,33,31,20)(18,32,38,27)(25,36,28,34)(26,35,40,37);(1,2,4,6)(3,8,7,5)(10,17)(12,15)(13,31)(14,30)(18,34)(19,33)(20,22)(21,23)(24,35)(25,28)(26,27)(32,40)(36,38)(37,39);(9,11)(10,18)(12,25)(13,32)(14,26)(15,28)(16,29)(17,34)(19,35)(20,39)(21,36)(22,37)(23,38)(24,33)(27,30)(31,40);(9,12,16,15)(10,19,23,22)(11,25,29,28)(13,14,30,31)(17,20,21,33)(18,35,38,37)(24,34,39,36)(26,27,40,32)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396246);