Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#454 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,19)(12,23)(13,18)(14,16)(15,30)(17,21)(20,27)(22,32)(24,35)(25,36)(26,33)(28,34)(29,37)(31,40)(38,39);(2,5)(6,7)(9,11,13,14)(10,16,18,19)(12,24,25,39)(15,29,27,22)(17,33,34,40)(20,37,30,32)(21,31,28,26)(23,38,36,35);(9,12)(10,17)(11,21)(13,25)(14,28)(15,31)(16,23)(18,34)(19,36)(20,38)(22,39)(24,29)(26,27)(30,35)(32,40)(33,37);(1,3,4,8)(2,6,5,7)(10,18)(16,19)(17,34)(20,30)(23,36)(32,37)(33,40)(35,38)Polycyclic presentation:
2^9|b^2=i,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407712);